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A 



TEXT BOOK 



OF 



GEOMETRICAL DRAWING 



FOR THE USE OF 



Mtc[)anxtQ anh ©cljools, 

IN WHICH 

THE DEFINITIONS AND RULES OF GEOMETRY ARE FAMILIARLY EXPLAINED, THE PRACTICAL PROB- 
LEMS ARE ARRANGED FROM THE MOST SIMPLE TO THE MORE COMPLEX, AND IN THEIR 
DESCRIPTION TECHNICALITIES ARE AVOIDED AS MUCH AS POSSIBLE; 

WITH ILLUSTRATIONS FOR DRAWING PLANS, SECTIONS AND ELEVATIONS OF 

BUILDINGS AND MACHINERY: 

AN 

INTRODUCTION TO ISOMETRICAL DRAWING, 

AND AN 

ESSAY ON LINEAR PERSPECTIVE AND SHADOWS: 

THE WHOLE ILLUSTRATED WITH 

FIFTY-SIX STEEL PLATES, 



B Y 

WM. MINIFIE, Architect, 

AND 
TEACHER OF DRAWING IN THE CENTRAL HIGH SCHOOL OF HALTIMOKE. 



PUBLISHED BY WI\r. MINIFIK & CO. 

NO. 114 B A L r I M O 11 K S T U E E T , 

BALTIMORE. 

1849. 



Entered, according to the Act of Congress, in the year 1849, 

BY WM. MINI FIE, 

In the Clerk's Office of the District Court of Maryland. 



STEREOTYPED AT THE 

BALTIMORE TYPE AND STEREOTYPE FOUNDRY. 

FIELDING LUCAS, JR., PROPRIETOR. 



JOHN D. TOY, PRINTER. 






o -a 



PREFACE. 



Having been for several years engaged in teaching Architectural and Me- 
chanical drawing, both in the High School of Baltimore and to private classes, 
I have endeavored without success, to procure a book that I could introduce 
as a text book; works on Geometry generally contain too much theory for 
the purpose, with an insufficient amount of practical problems ; and books on 
Architecture and Machinery are mostly too voluminous and costly, contain- 
ing much that is entirely unnecessary for the purpose. Under these circum- 
stances, I collected most of the useful practical problems in geometry from a 
variety of sources, simplified them and drew them on cards for the use of the 
classes, arranging them from the most easy to the more difficult, thus leading 
the students gradually forward; this was followed by the drawing of plans, 
sections, elevations and details of Buildings and Machinery, then followed 
Isometrical drawing, and the course was closed by the study of Linear per- 
spective and shadows ; the whole being illustrated by a series of short lectures 
to the private classes. 

I have been so well pleased with the results of this method of instruction, 
that I have endeavored to adopt its general features in the arrangement of 
the following work. The problems in constructive geometry have been selected 
with a view to their practical application in the every-day business of the 
Engineer, Architect and Artizan, while at the same time they afford a good 
series of lessons to facilitate the knowledge and use of the instruments requir- 
ed in mechanical drawing. 

The definitions and explanations have been given in as plain and simple 
language as the subject will admit of; many persons will no doubt think 
them too simple. Had the book been intended for the use of persons versed 
in geometry, very many of the explanations might have been dispensed with, 
but it is intended chiefly to be used as a first book in geometrical draiving, by 
persons who have not had the benefit of a mathematical education, and who 
in a majority of cases, have not the time or inclination to study any com- 
plex matter, or what is the same thing, that which may appear so to them. 
And if used in schools, its detailed explanations, we believe, will save time to 
the teacher, by permitting the scholar to obtain for himself much information 
that he would otherwise require to have explained to him. 

But it is also intended to be used for self-instruction^ without the aid of a 
teacher, to whom the student might refer for explanation of any dilliculty; 
under these circumstances I do not believe an explanation can be couched in 
too simple language. With a view of adapting the book to this class of stu- 
dents, the illustrations of each branch treated of, have l)('en made progressive, 
commencing with the plainest diagrams; and even in the nuire atlv;nu-eil, the 
object has been to instil principles rather than to produce clfect, as those once 



IV 

obtained, the student can either design for himself or copy from any subject 
at hand. It is hoped that this arrangement will induce many to study draw- 
ing who would not otherwise have attempted it, and thereby render them- 
selves much more capable of conducting any business, for it has been truly 
said by an eminent writer on Architecture, " that one workman is superior to 
another (other circumstances being the same) directly in proportion to his 
knowledge of drawing, and those who are ignorant of it must in many re- 
spects be subservient to others who have obtained that knowledge." 

The size of the work has imperceptibly increased far beyond my original 
design, which was to get it up in a cheap form with illustrations on wood, 
and to contain about two-thirds of the number in the present volume, but on 
examining some specimens of mathematical diagrams executed on wood, I 
was dissatisfied with their want of neatness, particularly as but few students 
aim to excel their copy. On determining to use steel illustrations I deemed 
it advisable to extend its scope until it has attained its present bulk, and even 
now I feel more disposed to increase than to curtail it, as it contains but few 
examples either in Architecture or Machinery. I trust, however, that the 
objector to its size will find it to contain but little that is absolutely useless to 
a student. 

In conclusion, I must warn my readers against an idea that I am sorry to 
find too prevalent, viz : that drawing requires but little time or study for its 
attainment, that it may be imbibed involuntarily as one would fragrance in a 
flower garden, with little or no exertion on the part of the recipient, not that 
the idea is expressed in so many words, but it is frequently manifested by their 
dissatisfaction at not being able to make a drawing in a few lessons as well 
as their teacher, even before they have had sufficient practice to have obtained 
a free use of the instruments. I have known many give up the study in con- 
sequence, who at the same time if they should be apprenticed to a carpenter, 
would be satisfied if they could use the jack plane with facility after several 
weeks practice, or be able to make a sash at the end of some years. 

Now this idea is fallacious, and calculated to do much injury; proficiency 
in no art can be obtained without attentive study and industrious persever- 
ance. Drawing is certainly not an exception ; but the difficulties will soon 
vanish if you commence with a determination to succeed ; let your motto be 
PERSEVERE, ucvcr Say "it is too diflScult;" you will not find it so difficult 
as you imagine if you will only give it proper attention ; and if my labors 
have helped to smooth those difficulties it will be to me a source of much 
gratification. 

WM. MINIFIE. 

Baltimore, 1st March, 1849. 



ILLUSTRATIONS. 



Definitions of lines and angles, 

Definitions of plane rectilinear superficies, . 

Definitions of the circle. 

To erect or let fall a perpendicular, 

Construction and division of angles. 

Construction of polygons, .... 

Problems relating to the circle, 

Parallel ruler, and its application. 

Scale of chords and plane scales, 

Protractor, its construction and application. 

Flat segments of circles and parabolas. 

Oval figures composed of arcs of circles. 

Cycloid and Epicycloid, 

Cube, its sections and surface. 

Prisms, square pyramid and their coverings, 

Pyramid, Cylinder, Cone and their surfaces, 

Sphere and covering and coverings of the regular Polyhedrons 

Cylinder and its sections, . 

Cone and its sections, .... 

Ellipsis and Hyperbola, .... 

Parabola and its application to Gothic arches, . 
To find the section of the segment of a cylinder through three 
points, ........ 

Coverings of hemispherical domes. 

Joints in circular and elliptic arches, .... 

Joints in Gothic arches, ...... 

Design for a Cottage — ground plan and elevation, 
Design for a Cottage — chamber plan and section, . 
Details of a Cottage — joists, roof and cornice, 
Details of a Cottage — parlor windows and plinth, 
Octagonal plan and elevation, ..... 

Circular plan and elevation, ..... 

Roman mouldings, ....... 

Grecian mouldings, ...... 

Plan, section and elevation of a wheel and pinion. 
To proportion the teeth of wheels, .... 

Cylinder of a locomotive, plan and section, 

Cylinder of a locomotive, transverse section and end view, 



given 



PLATE. 

i. 

ii. 
iii. 
iv. 

V. 

vi. 

vii. 

viii. 

ix. 

X. 

xi. 

xii. 

xiii. 

xiv. 

XV. 

xvi. 

xvii. 

xviii. 

xix. 

XX. 

xxi. 

xxii. 
. xxiii. 
xxiv. 

. XXV. 

xxvi. 

XX vii. 

xxviii. 

xxix. 

XXX, 

xxxi. 

xxxii. 

xxxiii. 

xxxiv. 

XXXV. 

xxxvi. 
xxxvii 



Vl 



PLATE. 

Isometrical cube, its construction, ..•••. xxxviii. 
Isometrical figures, triangle and square, ..... xxxix. 

Isometrical figures pierced and chamfered, ..... xl. 

Isometrical circle, method of describing and dividing it, . . . xli. 
Perspective — Visual angle, section of the eye, &c. .... xlii. 

Foreshortening and definitions of lines, . . . xliii. 

Squares, half distance, and plan of a room, . . . xliv. 

Tessellated pavements, ...... xlv. 

Square viewed diagonally. Circle. .... xlvi. 

Line of elevation, pillars and pjTamids, . . . xlvii. 

Arches parallel to the plane of the picture, . . . xlviii. 

Arches on a vanishing plane, ..... xlix. 

Application of the circle, ...... 1. 

Perspective plane and vanishing points, ... li. 

Cube viewed accidentally, ...... lii. 

Cottage viewed accidentally, ..... liii. 

Frontispiece. Street parallel to the middle visual ray, . liv. 
Shadows, rectangular and circular, ...... Iv. 

Shadows of steps and cylinder, ....... Ivi. 



a 



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PRACTICAL GEOMETRY. 



PLATE I. 

DEFINITIONS OF LINES AND ANGLES. 



1. A Point is said to have position without magnitude; and it is 
therefore generally represented to the eye by a small dot^ as at A. 

2. A Line is considered as length without breadth or thickness^ 
it is in fact a succession of points; its extremities therefore^ are 
points. Lines are of three kinds ; right lineSy curved lines, and 
mixed lines. 

3. A Right Line^ or as it is more commonly called^ a straight 
line, is the shortest that can be drawn between two given points 
as B. 

4. A Curve or Curved Line is that which does not lie evenly 
between its terminating points, and of which no portion, how- 
ever small, is straight ; it is therefore longer than a straight line 
connecting the same points. Curved lines are either regular or 
irregular. 

5. A Regular Curved Line, as C. is a portion of the circum- 
ference of a circle, the degree of curvature being the same 
throughout its entire * length. An irregular curved line has 
not the same degree of curvature throughout, but varies at dif- 
ferent points. 

6. A Waved Line may be either regular or irregular ; it is com- 
posed of curves bent in contrary directions. £^ is a regular 
waved line, the inflections on either side of the dotted line being 
equal ; a waved line is also called a line of double curvature of 
contrary flexure, and a serpentine line. 

7. Mixed Lines are composed of straight and curved lines, as D. 

8. Parallel Lines are those which have no inclination to tach 
other, as F, being every where equidistant; consequently they 
could never meet, though produced to infinity. 



8 



PLATE I. 



If the parallel lines G were produced^ they would form two 
concentric circles, viz: circles which have a common centre, 
whose boundaries are every where parallel and equidistant. 

9. Ijvclined Lines, as H and /, if produced, would meet in a 
point as at jfiT, forming an angle of which the point K is called 
the vertex or angular point, and the lines H and / the legs or 
sides of the angle K; the point of meeting is also called the 
summit of an angle. 

10. Perpendicular Lines. — Lines are perpendicular to each 
other when the angles on either side of the point of junction are 
equal; thus the lines Jf, 0. P are perpendicular to the line L 
M. The lines JV. 0. P are called also vertical lines and plumb 
lines, because they are parallel with any line to which a plummet 
is suspended ; the line Z. Jli" is a horizontal or level line ; hnes 
so called are always perpendicular to a plumb line. 

11. Vertical and Horizontal Lines are always perpendicular 
to each other, but perpendicular lines are not always vertical and 
horizontal; they may be at any inclination to the horizon pro- 
vided that the angles on either side of the point of intersection 
are equal, as for example the lines X. Y and Z. 

12. Angles. — Two right lines drawn from the same point, di- 
verging from each other, form an angle, as the lines S, Q. R. 
An angle is commonly designated by three letters, and the letter 
designating the point of divergence, which in this case is Q, is 
always placed in the middle. Angles are either acute, right or 
obtuse. If the legs of an angle are perpendicular to each other, 
they form a right angle as T. Q. i?, (mechanics' squares, if true, 
are always right angled;) if the sides are nearer together, as S. 
Q. i?, they form an acute angle ; if the sides are wider apart, or 
diverge from each other more than a right angle, they form an 
obtuse angle, as F. Q. jR. 

The magnitude of an angle does not depend on the length of 
the sides, but upon their divergence from each other ; an angle is 
said to be greater or less than another as the divergence is greater 
or less ; thus the obtuse angle V. Q. R is greater, and the acute 
angle S. Q. R is less than the right angle T, Q, R, 



Flat& 1. 



DEFIXITWXS OF LINES JXD AXGLES. 





H. 




H"" Nil /■ 



'^ 



!^' 



riate 2. 
DEFINITIONS. PLANE EECTILINEAR SUPERFICIES. 



TRIANGLES OR TPaGONS. 




QIMJJPJRATERALS. OJJARPiANGLES OR TETRAGONS. 



PARALLELOGPiAMS- . 




ir^Mirafio,. 



JUmaTikSoTis 



PLATE IL 

PLANE RECTILINEAR SUPERFICIES. 



13. A Superficies or Surface is considered as an extension of 
length and breadth without thickness. 

14. A Plane Superficies is an enclosed flat surface that will 
coincide in every place with a straight line. It is a succession of 
straight lines^ or to be more explicit^ if a perfectly straight edged 
ruler be placed on a plane superficies in any direction, it would 
touch it in every part of its entire length. 

15. When surfaces are bounded by right lines, they are said to be 
Rectilinear or Rectilineal. As all the figures on plate 
second agree with the above definitions, they are Plane Recti- 
linear Superficies. 

16. Figures bounded by more than four right lines are called 
Polygons; the boundary of a polygon is called its Perimeter. 

17. When Surfaces are bounded by three right lines, they are 
called Triangles or Trigons. 

18. An Equilateral Triangle has all its sides of equal length, 
and all its angles equal, as A. 

19. An Isosceles Triangle has two of its sides and two of its 
angles equal, as B, 

20. A Scalene Triangle has all its sides and angles unequal, 
as C. 

21. An Acute Angled Triangle has all its angles acute, as A 
and B. 

22. A Right Angled Triangle has one right angle; the side 
opposite the right angle is called the hypothenusc; the other sides 
are called respectively the base and perpendicular. The figures 
A. B, C, are each divided into two right angled triangles by tlie 
dotted lines running across them. 

23. An Obtuse Angled Triangle has one obtuse angle, as C. 

24. If figures A and B were cut out and folded on the dotted 
line in the centre of each, the opposite sides would exactly coin- 
cide ; they are therefore, regular triangles. 

25. Any of the sides of an equilateral or scalene triangle may be 
called its Base, but in the Isosceles triangle the side which is 



10 



PLATE II. 



unequal is so called^ the angle opposite the base is called the 
Vertex. 

26. The Altitude of a Triangle is the length of a perpendicular 
let fall from its vertex to its base^ as a. A. and h. B, or to its base 
extended^ as d. d, figure C. 

The superficial contents of a Triangle may be obtained by mul- 
tiplying the altitude by one half the base. 

27. When surfaces are bounded by four right lines^ they are called 
Quadrilaterals^ Quadrangles or Tetragons; either of the 
figures D. E. F. G. H and K may be called by either of those 
terms^ which are common to all four-sided right lined figures^ 
although each has its own proper name. 

28. When a Quadrilateral has its opposite sides parallel to each 
other^ it is called a Parallelogram ; therefore figures JD. E. F 
and G are parallelograms. 

29. When all the angles of a Tetragon are right angles^ the figure 
is called a Rectangle^ as figures D and E. 

If two opposite angles of a Tetragon are right angles^ the others 
are necessarily right too. 

30. If the sides of a Rectangle are all of equal lengthy the figure 
is called a Square^ as figure D, 

31. If the sides of a Rectangle are not all of equal lengthy two of 
its sides being longer than the others^ as figure E^ it is called an 
Oblong. 

32. When the sides of a parallelogram are all equal^ and the an- 
gles not right angles, two being acute and the others obtuse^ as 
figure -F, it is called a Rhomb^ or Rhombus ; it is also called a 
Diamond^ and sometimes a Lozenge^ more particularly so when 
the figure is used in heraldry. 

33. A parallelogram w^hose angles are not right angles^ but whose 
opposite sides are equal^ as figure G, is called a Rhomboid. 

34. If two of the sides of a Quadrilateral are parallel to each 
other as the sides H and in ^g. H^ it is called a Trapezoid. 

35. All other Quadrangles are called Trapeziums^ the term being 
applied to all Tetragons that have no two sides parallel^ as K. 



Note. The terms Trapezoid and Trapezium are applied indiscrimin- 
ately by some writers to either of the figures H and K; by others, fig. H is 
called a Trapezium and fig. K a Trapezoid^ and this appears to be the more 
correct method; but as Trapezoid is a word of comparatively modern origin, 
I have used it as it is most generally applied by modern writers, more par- 
ticularly so in works on Architecture and Mechanics. 



PLATE III, 



11 



36. A Diagonal is a line running across a Quadrangle^ connect- 
ing its opposite corners^ as the dotted lines in figs. D and F. 

Note. — I have often seen persons who have not studied Geometry, much 
confused in consequence of the number of names given to the same figure, 
as for example fig. D. 

1st. It is a plane Figure — see paragraph 14. 

2nd. It is Rectilineal^ being composed of right lines. 

3rd. It Ls a Quadrilateral, being composed of four lines. 

4th. It is a Quadrangle, having four angles. 

5th. It is a Tetragon, having four sides. 

6th. It is a Parallelogram, its opposite sides being parallel. 

7th. It is a Rectangle, all its angles being right angles. 

All the above may be called common names, because they are applied to all 
figures having the same properties. 

8th. It is a Square, which is its proper name, distinguishing it from all other 
figures, to which some or all of the above terms may be applied. 

All of them except 7 and 8, may also be applied to fig. F, with the same 
propriety as to fig. D ; besides these, fig. F has four proper names distin- 
guishing it from all other figures, viz : a Rhomb, Rhombus, Diamond and 
Lozenge. 

If the student will analyze all the other figures in the same manner, he 
will soon become perfectly familiar with them, and each term will convey to 
his mind a clear definite idea. 



PLATE III. 

DEFINITIONS OF THE CIRCLE. 

1st. A Circle is a plane figure bounded by one curve line, 
every where equidistant from its centre, as fig. 1. 

2nd. The boundary line is called the Circumference or Pe- 
riphery, it is also for convenience called a Circle. 

3rd. The Centre of a circle is a point within the circumference, 
equally distant from every point in it, as C, fig. 1 . 

4th. The Radius of a circle is a line drawn from the centre to 
any point in the circumference, as C. Jl, (\ /> or ('. 1), lig. 1. 

The plural of Radius is Radti. All radii of the same circle are 
of equal lenii;th. 

5th. The Diainiictek of a circle is any rii;ht line drawn through 
the centre to opposite points of the circumferenci^, as .7. B, fii^. 1 



J 



12 



PLATE III. 



The length of the diameter is equal to two radii; there may 
be an infinite number of diameters in the same circle^ but they 
are all equal. 

6th. A Semicircle is the half of a circle^ as fig. 2; it is bounded 
by half the circumference and by a diameter. 

7th. A Segment of a circle is any part of the surface cut off 
by a right line^ as in fig. 3. Segments may be therefore greater 
or less than a semicircle. 

8th. An Arc of a circle is any portion of the circumference cut 
off, as C. G, D or E. G. F, fig. 3. 

9th. A Chord is a right line joining the extremities of an arc^ 
as C. D and E, F, fig. 3. The diameter is the chord of a 
semicircle. The chord is also called the Subtense. 

10th. A Sector of a circle is a space contained between two 
radii and the arc which they intercept^ as E, C. Dj or 0. C. 
H, fig. 4. 

11th. A Quadrant is a sector whose area is equal to one-fourth 
of the circle^ as fig. 5; the arc D. E being equal to one-fourth 
of the whole circumference^ and the radii at right angles to 
each other. 

12. A Degree. — The circumference of a circle is considered as 
divided into 360 equal parts called Degrees, (marked °) each 
degree is divided into 60 minutes (marked ') and each minute 
into 60 seconds (marked ")] thus if the circle be large or small, 
the number of divisions is always the same, a degree being equal 
to 1 -360th part of the whole circumference, the semicircle equal 
to 180°, and the quadrant equal to 90°. The radii drawn from 
the centre of a circle to the extremities of a quadrant are always 
at right angles to each other; a right angle is therefore called an 
angle of 90°. If we bisect a right angle by a right line, it would 
divide the arc of the quadrant also into two equal parts, each part 
equal to one-eighth of the whole circumference containing 45°; 
if the right angle were divided into three equal parts by straight 
fines, it would divide the arc into three equal parts, each containing 
30°. Thus the degrees of the circle are used to measure angles, 
and when we speak of an angle of any number of degrees, it is 
understood, that if a circle with any length of radius^ he struck 
with one foot of the dividers in the angular point, the sides of the 
angle will intercept a portion of the circle equal to the number of 
degrees given. 

Note. — This division of the circle is purely arbitrary, but it has existed 



riMe 3: 
DEFINITIONS OF THE CIP^CLE. 




r, A 



2. 
S'EJyaCIRCLE. 



SEGMENTS. 
(i 





QUADRANT. 



COMTLEMENT. 
E 





SUETI.EMEST. 




(I 
TANCESTAr 



11 


--, ^ 


(' 


-JV 


f 

r. 



SINE. 




Z. 



II 





IV" yitnfir 



PLATE III. 



13 



from the most ancient times and every where. During the revolutionary 
period of 1789 in France, it was proposed to adopt a decimal division, by 
which the circumference was reckoned at 400 grades ; but this method was 
never extensively adopted and is now virtually abandoned. 

13. The Complement of an Arc or of an Jingle is the difference 
between that arc or angle and a quadrant; thus E. D fig. 6 is the 
complement of the arc D. By and E. C. D the complement of 
the angle D. C. B, 

14. The Supplement of an Arc or of an Angle is the difference 
between that arc or angle and a semicircle; thus D. A fig. 7, is 
the supplement of the arc D, B, and D. C, A the supplement 
of the angle B, C, D. 

15. A Tangent is a right line^ drawn without a circle touching it 
only at one point as B, E fig. 8 ; the point v/here it touches the 
circle is called the point of contact, or the tangent point. 

16. A Secant is a right line drawn from the centre of a circle 
cutting its circumference and prolonged to meet a tangent as 
C. E fig. 8. 

Note. — Secant Point is the same as jpoint of intersection, being the 
point where two lines cross or cut each other. 

17. The Co-Tangent of an arc is the tangent of the comple- 
ment of that arc, as H, K fig. S. 

Note. — The shaded parts in these diagrams are the given angles, but 
if in fig. 8, D. C. H be the given angle and D. H the given arc, then H. K. 
would be the tangent and B. E the co-tangent. 

18. The Sine of an arc is a line drawn from one extremity, per- 
pendicular to a radius drawn to the other extremity of the arc as 
D. F fig. 9. 

19. The Co-sine of an arc is the sine of the complement of that 
arc as L. D fig. 10. 

20. The Versed Sine of an arc is that part of the radius inter- 
cepted between the sine and the circumference as F, B fig. 9. 

21. In figure 11, we have the whole of the foregoing definitions 
illustrated in one diagram. C. H — C. D — C. B and C. A are 
Radii; A, B the Diameter; B, C, D a. Sector; B. C. //a 
Quadrant. Let B, C. D be the given Angle, and B. D 
the given Arc, then B. D is the Chord, D, H the Complement, 
and D. A the Supplement of the arc; D. C. H the Complement 
and D, C. A the Supplement of the given angle; B. E the Tan- 
gent and //. K the Co-tangent, C. E the Secant and C, K the 
Co-secant, F. D the Sine, L. D the Co-sine and F. B the 
Versed Sine, 



14 



PLATE IV. 

TO ERECT OR LET FALL A PERPENDICULAR. 



Problem L Figure L 



To bisect the right line A. B by a perpendicular. 

1st. With any radius greater than one half of the given line^ 
and with one point of the dividers in A and B successively^ 
draw two arcs intersecting each other^ in C and D. 

2nd. Through the points of intersection draw C. D^ which is 
the perpendicular required. 



Problem 2. Fig. 2. 



From the point D in the line E. F ^o erect a perpendicular. 

1st. With one foot of the dividers placed in the given point D 
with any radius less than one half of the line^ describe an arc^ 
cutting the given line in B and C. 

2nd. From the points B and C with any radius greater than B. 
D, describe two arcs^ cutting each other in G. 

3rd. From the point of intersection draw G. D, which is the per- 
pendicular required. 



Problem 3. Fig. 3. 



To erect a perpendicular when the point D is at or near the end of 

a line. 

1st. With one foot of the dividers in the given point D with 
any radius^ as D. E, draw an indefinite arc G. H. 

2nd. With the same radius and the dividers in any point of the 
arc^ as E^ draw the arc B. D. F^ cutting the line C. D in B. 

3rd. From the point B through E draw a right line^ cutting 
the arc in F. 



Flate 4. 
TO EPy^ECT cm LET FALL A PFPiFEXDICULAPL. 



Fiq. 1 


(' 






I): 


F 



Fiti 5. 



Fui 4. 



Fu,. :' 



Fiti .i 



Fill. 6 



\]i 



A 









D 






Fia /. 








. I- 




M 










Or 












:K 



B 11 



Fiq 8. 



(■ B 



7.^--2'lTnine . 



niman-kSori: 



PLATE IV. 



15 



4th. From F draw F. D, which is the perpendicular required. 

]^oTE. — It will be perceived that the arc B. D. F is a semicircle, and 
the right line B. F di diameter ; if from the extremities of a semicircle right 
lines be drawn to any point in the curve, the angle formed by them will be 
a right angle. This affords a ready method for forming a " square corner^'* 
and will be found useful on many occasions, as its accuracy may be de- 
pended on. 



Problem 4. Fig. 4. 



Another method of erecting a perpendicular ivhen at or near the 

end of the line. 

Continue the hne H. D toward C, and proceed as in problem 
2; the letters of reference are the same. 



Problem 5. Fig. 5. 



From the point D to let fall a perpendicular to the line A. B. 

1st. With any radius greater than D. G and one foot of the com- 
passes in D, describe an arc cutting A. B m E and F. 

2nd. From E and F with any radius greater than E. G, describe 
two arcs cutting each other as in C. 

3rd. From D draw the right line D. C, then D. G is the per- 
pendicular required. 



Problem 6. Fig. 6. 



When the point D is nearly opposite the end of the line, 

1st. From the given point D, draw a right line to any point of the 

line A. B as 0. 
2nd. Bisect 0. D by problem 1^ in E. 
3rd. With one foot of the compasses in E with a radius equal to 

E. D or E. describe an arc cutting A. B in F. 
4th. Draw D. F which is the perpendicular required. 

Note. — The reader will perceive that we have arrived at the same result 
as we did by problem 3, but by a different process, the right angle being 
formed within a semicircle. 



16 



PLATE IV. 



Problem 7. Fig. 7. 



Another method of letting fall a perpendicular when the given point 
D is nearly opposite the end of the line. 

1st. With any radius as F. D and one foot of the compasses in 
the line Jl. B as at F^ draw an arc D. H. C. 

2nd. With any other radius as E. D draw another arc D. K. 
C, cutting the first arc in C and D. 

3rd. From D draw D, C, then D, G is the perpendicular re- 
quired. 

Note. — The points E and F from which the arcs are drawn, should be 
as far apart as the line A. B will admit of, as the exact points of intersec- 
tion can be more easily found, for it is evident, that the nearer two lines cross 
each other at a right angle, the finer will be the point of contact. 

Problem 8. Fig. 8. 



To erect a perpendicular at D the end of the line CD. with a scale 

of equal parts, 

1st. From any scale of equal parts take three in your dividers^ 

and with one foot in D^ cut the line C. D in B, 
2nd. From the same scale take four parts in your dividers^ and 

with one foot in D draw an indefinite arc toward E. 
3rd. With a radius equal to five of the same parts^ and one foot 

of the dividers in B^ cut the other arc in E. 
4th. From E draw E. Dy which is the perpendicular required. 
Note 1st. If four parts were first taken in the dividers and laid off on 

the line C D, then three parts should be used for striking the indefinite 

arc, at A, and the five parts struck from the point C, which would give the 

intersection A, and arrive at the same result. 
2nd. On referring to the definitions of angles, it will be found that the side 

of a right angled triangle opposite the right angle is called the Hypothe- 

nuse; thus the line E. B is the hypothenuse of the triangle E. D. B. 
3rd. The square of the hypothenuse of a right angled triangle is equal to 

the sum of the squares of both the other sides. 
4th. The square of a number is the product of that number multiplied by 

itself. 
Example. The length of the side D. E is 4, which multiplied by 4, wiU 

give for its square 16. The length of D. B is 3, which multiplied by 3, gives 

for the square 9. The products of the two sides added together give 25. 

The length of the hypothenuse is 5, which multiplied by 5, gives also 25. 
5th. The results will always be the same, but if fractional parts are used in 



PLATE IV. 



17 



the measures, the proof is not so obvious, as the multiplication would be 

more complicated. 
6th. 3, 4 and 5 are the least whole numbers that can be used in laying down 

this diagram, but any multiple of these numbers may be used ; thus, if we 

multiply them by 2, it would give 6, 8 and 10 ; if by 3, it would give 9, 12 

and 15 ; if by 4 — 12, 16 and 20, and so on. The greater the distances 

employed, other things being equal, the greater w^ill be the probable accuracy 

of the result. 
7th. We have used a scale of equal parts without designating the unit of 

measurement, which may be an inch, foot, yard, or any other measure. 
8th. As this problem is frequently used by practical men in laying .off w^ork, 

we will give an illustration. 
Example. Suppose the line C. D to be the front of a house, and it is desired 

to lay off the side at right angles to it from the corner D. 
1st. Drive in a small stake at D, put the ring of a tape measure on it and 

lay off twelve feet toward B. 
2nd. With a distance of sixteen feet, the ring remaining at D, trace a short 

circle on the ground at E. 
3rd. Remove the ring to J5, and with a distance of twenty feet cut the first 

circle at E. 
4th. Stretch a line from D to E, which will give the required side of the 

building. 



PLATE V. 



CONSTRUCTION AND DIVISION OF ANGLES. 



Problem 9. Fig. 1. 



The length of thC' sides of a Triangle A. B., C. D. and E. F 
being given, to construct the Triangle, the two longest sides to be 
joined together at A. 

1st. With the length of the hne C. D for a radius and one foot 

in J?, draw an arc at G. 
2nd. With the length of the line E. F for a radius and one foot 

in By draw an arc cutting the other arc at G, 
3rd. From the point of intersection draw G. •// and G. B^ which 

complete the figure. 



18 PLATE V. 

Problem 10. Fig. 2. 



To construct an Jingle at K equal to the Jingle H. 

1st. From Hwith any radius, draw an arc cutting the sides of the 

angle as at JVf. JY. 
2nd. From K with the same radius, describe an indefinite arc 

at 0. 
3rd. Draw K, parallel to H. M. 

4th. Take the distance from J\I to JY and apply it from to P. 
5th. Through P draw K, P, which completes the figure. 



Problem U. Fig. 3. 



To Bisect the given Jingle Q by a Right Line, 

1st. With any radius and one foot of the dividers in Q draw an 

arc cutting the sides of the angle as in R and *S'. 
2nd. With the same or any other radius, greater than one half 

R. S^ from the points S and P, describe two arcs cutting each 

other, as at T. 
3rd. Draw T. Q, which divides the angle equally. 

Note. — This problem may be very usefully applied by workmen on many 
occasions. Suppose, for example, the corner Q be the corner of a room, 
and a washboard or cornice has to be fitted around it; first, apply the bevel 
to the angle and lay it down on a piece of board, bisect the angle as above, 
then set the bevel to the centreline, and you have the exact angle for cutting 
the mitre. This rule will apply equally to the internal or external angle. 
Most good practical workmen have several means for getting the cut of the 
mitre, and to them this demonstration will appear unnecessary, but I have 
seen many men make sad blunders, for want of knowing this simple rule. 

Problem 12. Fig. 4. 



To Trisect a Right Angle, 

1st. From the angular point V with any radius, describe an arc 
cutting the sides of the angle, as in X and TV, 

2nd. With the same radius from the points X and Wy cut the arc 
in Fand Z. 

3rd. Draw F. V and Z, F, which will divide the angle as re- 
quired. 




/f "' Miitillf. 



PLATE V. 



19 



Problem 13. Fig. 5, 



In the triangle A. B. C^ to describe a Circle touching all its sides, 

1st. Bisect two of the angles by problem 11^ as J^ and B, the 

dividing lines will cut each other in D, then D is the centre of 

the circle. 
2nd. From D let fall a perpendicular to either of the sides as at 

Fj then D, F is the radius^ with which to describe the circle 

from the point D. 



Problem 14. Fig. 6. 



071 the given line A. B to construct an Equilateral Triangle, the 
line A. B to be one of its sides, 

1st With a radius equal to the given line from the points A and 
B, draw two arcs intersecting each other in C, 
2nd. From C, draw C. A and C, B, to complete the figure. 



PLATE VI. 



CONSTRUCTION OF POLYGONS 



A polygon of 3 sides is called a Trigon. 
4 " 



u 
a 
cc 
i( 

(C 

cc 



5 

6 

7 

8 

9 

10 

11 

12 



cc 

cc 

cc 

cc 

cc 

cc 

cc 



cc 
cc 

(C 

cc 
cc 
cc 
(( 

(C 

cc 



Tetragon. 



Pentagon. 

Hexagon. 

Heptagon. 

Octagon. 

Enneagon or Nonagon. 

Decaiifon. 

UndecaiJon. 

Dodeca^jfon. 



1st. When the sides of a polygon are all of ecjual length and all the 
angles are e(|ual/it is called a regular polygon; if une(iual, it is 
called an irregular polygon. 



20 PLATE VI. 

2nd. It is not necessary to say a regular Hexagon^ regular Octa- 
gon^ &LC. ; as when either of those figures is named^ it is always 
supposed to be regular^ unless otherwise stated. 



Problem 15. Fig. 1. 



On a given line A. B ^o construct a square whose side shall be equal 

to the given line. 

1st. With the length A. B for a radius from the points J. and B, 

describe two arcs cutting each other in C, 
2nd. Bisect the arc C. J. ov C. B in D. 
3rd. From C^ with a radius equal to C. D^ cut the arc B.E inE 

and the arc A. F in F, 
4th. Draw ^. E^ E. F and F, By which complete the square. 



Problem 16. Fig. 2. 



In the given square G. H. K. J^ fo inscribe an Octagon. 

1st. Draw the diagonals G. K and H. J^ intersecting each other 
in P. 

2nd. With a radius equal to half the diagonal from the corners 
G. H, K and J, draw arcs cutting the sides of the square in 0. 
0. 0, &:c. 

3rd. Draw the right lines 0. 0., 0. 0^ &c.^ and they will com- 
plete the octagon. 

This mode is used by workmen when they desire to make a 
piece of wood round for a roller^ or any other purpose ; it is first 
made square^ and the diagonals drawn across the end ; the dis- 
tance of one-half the diagonal is then set ofF^ as from G to i? in 
the diagram^ and a guage set from H to R which run on all the 
corners^ gives the lines for reducing the square to an octagon ; 
the corners are again taken ofF^ and finally finished with a tool 
appropriate to the purpose. The centre of each face of the octa- 
gon gives a line in the circumference of the circle^ running the 
whole length of the piece; and as there are eight of those lines 
equidistant from each other, the further steps in the process are 
rendered very simple. 



Flate 0. 



rOXSTRUCTION OF POLYGONS. 



Fiq. I. 





Fin. y. 




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PLATE VI. 



21 



Problem 17. Fig. 3. 



In a given circle to inscribe an Equilateral Triangle^ a Hexagon 

and Dodecagon, 

1st. For the Triangle^ with the radius of the given circle from 
any point in the circumference^ as at Jly describe an arc cutting 
the circle in B and C, 

2nd. Draw the right hne B, C, and with a radius equal B. C, from 
the points B and C, cut the circle in D. 

3rd. Draw D. B and D. C, which complete the triangle. 

4th. For the Hexagon^ take the radius of the given circle and 
carry it round on the circumference six times, it will give the 
points J. B. E. D. F, C, through them, draw the sides of the 
hexagon. The radius of a circle is always equal to the side of 
an hexagon inscribed therein. 

5th. For the Dodecagon, bisect the arcs between the points 
found for the hexagon, which will give the points for inscribing 
the dodecagon. 

Problem 18. Fig. 4. 



In a given Circle to inscribe a Square and an Octagon, 

1st. Draw a diameter ^. B, and bisect it with a perpendicular 
by problem 1, giving the points C. D. 

2nd. From the points ,A. C. B. D, draw the right lines forming 
the sides of the square required. 

3rd. For the Octagon, bisect the sides of the square and draw 
perpendiculars to the circle, or bisect the arcs between the points 
Jl. C. B, Z>, which will give the other angular points of the re- 
quired octagon. 



Problem 19. Fig. 5. 



On the given line O. P to construct a Pentagon^ O. P being ike 

length of the side. 

1st. With the length of the line 0. P from 0, describe the semi- 
circle P. Q, meeting the line P. 0, extended in Q. 

2nd. Divide the semicircle into {^\e equal parts and from draw 
lines through the divisions 1, 2 and 3. 



22 PLATE VI. 

3rd. With the length of the given side from P^ cut \ m S^ from 
*S cut 2^ in R^ and from Q cut 2m R; connect the points 0. 
Q. R, S. P by right Hnes^ and the pentagon will be complete. 

Problem 20. Fig. 6. 



On the given line A. B ^o construct a Heptagon ^ A. B being the 

length of the side. 

1st. From A with A, B for a radius, draw the semicircle B, H, 

2d. Divide the semicircle into seven equal parts, and from A 
through 1, 2, 3, 4 and 5, draw indefinite hues. 

3rd. From B cut the Hne ^ 1 in C, from G cut ^ 4 in jP, from 
F cut ^ 3 in £*, and from C cut ^ 2 in D, connect the points by 
right lines and complete the figure. 

Any polygon may be constructed by this method. The rule 
is, to divide the semicircle into as many equal parts as there 
are sides in the required polygon, draw lines through all the 
divisions except two, and proceed as above. 

Considerable care is required to draw these figures accurately, 
on account of the difliculty of finding the exact points of inter- 
section. They should be practised on a much larger scale. 



PLATE VII. 

Problem 21. Fig. 1. 



To find the Centre of a Circle. 

1st. Draw any chord, as A. jB, and bisect it by a perpendicular E. 

D, which is a diameter of the circle. 
2nd. Bisect the perpendicular E, D by problem 1, the point of 

intersection is the centre of the circle. 

Figure 2. 



Another method of finding the Centre of a Circle, 

1st. Join any three points in the circumference as F, G, H. 
2nd. Bisect the chords F, G and G, H by perpendiculars, their 
point of intersection at C is the centre required. 



rian- 7. 



r/KfliJJlMS lU'lLATLyc. TO TIU'l llUrL]]. 



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PLATE VII. 



23 



Problem 22. Fig. 3. 



To draw a Circle through any three points not in a straight line, 

as M. N. O. 

1st. Connect the points by straight lines^ which will be chords to 
the required circle. 

2nd. Bisect the chords by perpendiculars^ their point of inter- 
section at C is the centre of the required circle. 

3rd. With one foot of the dividers at C, and a radius equal to 
C, M, C. JV*, or C, 0; describe the circle. 

Problem 23. Fig. 4. 



To find the Centre for describing the Segment of a Circle. 

1st. Let P. R be the chord of the segment^ and P. S the rise. 

2nd. Draw the chords P. Q and Q. P; and bisect them by per- 
pendiculars ; the point of intersection at (7, is the centre for 
describing the segment. 



Problem 24. Fig. 5. 



To find a Right Line nearly equal to an Arc of a Circle^ as H. I. K. 

1st. Draw the chord K. K^ and extend it indefinitely toward 0. 
2nd. Bisect the segment in I^ and draw the chords H. I and /. K. 
3rd. With one foot of the dividers in H, and a radius equal to H. 

ly cut H. in My then with the same radius, and one foot in My 

cut H. again in JV. 
4th. Divide the difference K. JV into three equal parts, and extend 

one of them toward 0, then will the right line H, be nearly 

equal to the curved line H. I. K. 

Problem 25. Fig. 6. 



To find a Right Line nearly equal to the Semicircumfcrcnce A. F. B. 

1st. Draw the diameter A. B, and bisect it by the perpendicular 

F. H; extend P. // indefinitely toward G. 
2nd. Divide the radius C. H into four equal parts, and extend 

three of those parts to G. 
3rd. At F draw an indefinite right line D. E. 



24 



PLATE VII. 



4th. From G through J, the end of the diameter J. B, draw G, 
J. D, cutting the hne D, E in D, and from G through B draw 
G, B. E, cutting D, E in E, then will the line D, E be nearly 
equal to the semicircumference of the circle^ and the triangles 
I). G. E and J. G. B will be equilateral. 

Note. — The right lines found by problems 24 and 25, are not mathemati- 
cally equal to the respective curves, but are sufficiently correct for all 
practical purposes. Workmen are in the habit of using the following 
method for finding the length of a curved line : — 

They open their compasses to a small distance, and commencing at one 
end, step off the whole curve, noting the number of steps required, and the 
remainder less than a step, if any; they then step off the same number of 
times, with the same distance on the article to be bent around it, and add 
the remainder, which gives them a length sufficiently true for their purpose : 
the error in this method amounts to the sum of the differences between the 
arc cut off by each step, and its chord. 



PLATE VIII. 

PARALLEL RULER AND APPLICATION. 



Figure 1. 



The parallel ruler figured in the plate consists of two bars of wood 
or metal ^. B and C. D, of equal lengthy breadth and thickness^ 
connected together by two arms of equal length placed diagonal- 
ly across the bars^ both at the same angle^ and moving freely on 
the rivets which connect them to the bars; if the bar ^. B be 
kept firmly in any position and the bar C. D moved^ the ends of 
the arms connected to C. D will describe arcs of circles and 
recede from A. B until the arms are at right angles to the bars^ 
as shewn by the dotted lines; if moved farther round^ the bars will 
again approach each other on the other side. 

The bars of which' the ruler is composed^ being parallel to each 
other, it follows^ that if either edge of the instrument is placed 
parallel with a line and held in that position^ another line may be 
drawn parallel to the first at any distance within the range of the 
instrument. This is its most obvious use; it is generally apphed 
to the drawing of inclined parallel lines in mechanical drawings, 
vertical and horizontal lines being more easily drawn with the 
square, when the drawing is attached to a drawing board. 



Flate 8 . 
PARALLEL RULER . 

o.nd its Applix^ation 



FujJ. 




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Fig. 4. 



Mg.5. 




Yr^'-Mira/U 



L ixMZUK iX J J^. 



PLATE VIII. 



25 



Application. — Problem 26. Fig. 2. 



To divide the Line E. F into any number of equal parts ^ say 12. 

1st. From E draw E, G at any angle to E. F, and step off with 
any opening of the dividers twelve equal spaces on E. G. 

2nd. Join i^ 12, and with the parallel ruler draw lines through 
the points of division in E. G, parallel to 12 i^^ intersecting E. 
F, and dividing it as required. 



Problem 27. Fig. 3. 



To divide a Line of the length of G. H in the same proportion 
as the Line I. K is divided, 

1st. From / draw ahne at any angle and make /. L equal to G. H. 

2nd. Join the ends K and Z by a right line, and draw lines par- 
allel to it through all the points of division, to intersect /. L^ then 
/. L will be divided in the same proportion as /. K. 



Problem 28. Fig. 4. 



To reduce the Trapezium A. B. C. D to a Triangle of equal area. 

1st. Prolong C. D indefinitely. 

2nd. Draw the diagonal J.. D^ place one edge of the ruler on the 
line A. D and extend the other edge to J5, then draw B, E^ cut- 
ting C. D extended in E. 

3rd. Join A. Ey then the area of the triangle A. E, C will be equal 
to the trapezium A. B, C, D, 



Problem 29. Fig. 5. 



To reduce the irregular Pentagon F. G. H. I. K. to a Tetragon 
and to a Triangle ^ each of equal area ivith the J^cntagon. 

1st. Prolong /. H indefinitely. 

2nd. Draw the diagonal F. 7/ and G. J\/ parallel to it, cutting 7. // 

in M, and draw F. M. 
3rd. Prolong K. I indefinitely toward L, 
4th. Draw the diagonal F. I and draw M. L parallel to it, cutting 

K. Lm L, • 



26 



PLATE IX. 



5th. Draw F. L, then the triangle F, L. K, and the tetragon K. 
F. M. I, are equal in area to the given pentagon. 



PLATE IX. 

CONSTRUCTION OF THE SCALE QF CHORDS AND ITS APPLL 

CATION.— PLANE SCALES. 



Problem 30. Fig. 1. 



To Construct a Scale of Chords. 

Let ^^. B he sl rule on which to construct the scale. 

1st. With any radius^ and one foot in D, describe a quadrant; 

then draw the radii D. C and D. E. ' 
2nd. Divide the arc into three equal parts as follows : — With the 

radius of the quadrant^ and the dividers in C, cut the arc in 60 ; 

then^ with one foot in E^ cut the arc in 30. 
3rd. Divide these spaces each into three equal parts^ when the 

quadrant will be divided into nine equal parts of 10^ each. 
4th. From C, draw chords to each of the divisions, and transfer 

them^ as shewn by the dotted lines^ to ^. B, 
5th. Divide each of the divisions on the arc into ten equal parts^ 

and transfer the chords to A. B, when we shall have a scale of 

chords corresponding to the respective degrees. 

Note 1. — This scale is generally found on the plane scale which accom- 
panies a set of drawing instruments, and marked C, or Ch. 

Note 2. — Any scale of chords may be reconstructed by using the chord of 
60° as a radius for describing the quadrant. 



Application. — Problem 31. Fig. 2. 



To lay down an Jingle at ¥y of any number of degrees^ say 25, 
the line G. F to form one side of the Angle. 

1st. Take the chord of 60° in the dividers^ and with one foot in 
F^ describe an indefinite arc^ cutting G. F in H. 



^Mate 9. 



SOnLE OF CHORDS 



I'la. I 




SCALES OF EOrylL FA UTS. 



W 9 8 7 6 5 4 3 2 10 



0223456789 10 



J5 




/■'u/ / 



/'■/// 




PLATE IX. 



27 



2nd. From the scale take 25° in the dividers^ and with one foot 

in H^ cut the arc in K. 
3rd. Through K^ draw K. F^ which completes the required angle. 

If we desire an angle of 15° or 30°^ take the required number 

from the scale, and cut the arc in and P, and in the same 

manner for any other angle. 



Problem 32. Fig. 2. 



To measure an Angle F, already Laid down. 

1st. With the chord of 60°, and one foot of the dividers in the 
angular point, cut the sides of the angle in H and K. 

2nd. Take the distance H. K in the dividers, and apply it to the 
scale, which will shew the number of degrees subtended by the 
angle. 



SCALES OF EQUAL PARTS. 



1st. Scales of equal parts may be divided into two kinds, viz: — 
Those which consist of two or three lines, divided by short 
parallel lines, at right angles to the other, like fig. 3, or those 
which are composed of several parallel lines, divided by diagonal 
and vertical lines, like figs. 4 and 5 : the first kind are called 
simple scales^ the second diagonal scales. 

2nd. Scales of equal parts may be made of any size, and may be 
made to represent any unit of measure : thus each part of a scale 
may be an inch, or the tenth of an inch, or any other space, and 
may represent an inch, foot, yard, fathom, mile, or degree, or any 
other quantity. 

3rd. The measure which the scale is intended to represent, is 
called the tinit of meastirement. In architectural or mechanical 
drawings, the unit of measurement is generally a foot, which is 
subdivided into inches to correspond with the common foot rule. 
For working drawings, the scale is generally large. A very 
common mode of laying down working drawings, is to use a 
scale of one and a half inches to the foot ; tliis gives one-eighth 
of an inch to an inch, which is equal to one-eighth of the full 
size. This is very convenient, as every workman has a scale on 



28 



PLATE IX. 



his rule^ which he can apply to the drawing with facihty. Scales 
are generally made to suit each particular case^ dependant on the 
size of the object to be represented^ and on the size of the paper 
or board on w^hich the drawing is to be made. 

4th. To DRAW A Scale. If we have a definite size for each 
part of a required scale^ say one-quarter of an inch^ we have only 
to extend the dividers to that measure^ step off the parts and 
number them^ reserving the left hand space to be subdivided for 
inches. Care should be taken to have the scale true. It should 
be proved by taking two^ three^ four or five parts in the dividers^ 
and applying it to several parts of the scale ; when^ if found 
correct, the drawing may be proceeded with. It is much easier 
to draw another scale if the first is imperfect, than to correct a 
drawing made from a false scale. 

5th. Fig. 3 requires but little explanation. It is called a quarter 
of an inch scale, as each unit of the scale is a quarter of an inch ; 
the starting point of a scale marked 0, is called its Zero. The 
term is not very common among practical men, except when 
applied to the thermometer ; and, when w-e say the thermometer 
is down to Zero, we mean that it is at the commencement of the 
scale. It is better to number a scale above and below, as in the 
figure ; for, if we wish to take a measure of any number of feet 
less than 10, and inches, say 3 feet 6 inches, we place one foot 
of the dividers at 3, numbered from below, and extend the other 
out to 6 inches. If, on the other hand, we have to measure a 
number of feet more than 10, say 13, w^e should place one foot 
in the division marked 10, on the top of the scale without the 
plate, and extend it to 3, w^hen we are enabled to read the 
quantity at sight, without any mental operation, as we must do if 
the scale is only numbered as below. For example, to take 13 
feet, we must place one foot of the dividers at 20, and extend it 
to 7. This operation is simple, it is true, but it requires us to 
subtract 7 from 20 to get 13, instead of reading from the scale 
as we could do from the upper numbers. In taking a large space 
in the dividers, it is always better to take the w^hole numbers 
first, and add the inches or other fractions afterwards. The space 
on the left hand in this figure, is divided into twelve parts for 
inches. 



PLATE IX. 



29 



Figure 4. 



Is a HALF INCH DIAGONAL ScALE^ divided for feet and inches. 
To draw a scale of this kind : — 

1st. Draw 7 hnes parallel to each other^ and equidistant. 

2nd. Step off spaces of half an inch each^ and draw^ lines through 
the divisions across the whole of the spaces. 

3rd. Divide the top of the first space into two equal parts^ draw 
the diagonal lines^ and number them as in the diagram. 

To take any measure from this scale, say 2^ V^^ we must place the 
dividers on the first line above the bottom on the second division 
on the scale, and extend the other foot to the first diagonal line, 
numbered 1, which will give the required dimension. If we 
wish to take 2' 1 V^^ we must place one leg of the dividers the 
same as before, and extend the other to the second diagonal line, 
which gives the dimension. If we wish to take 1' 3^', we must 
place the dividers on the middle line in the first vertical division, 
and extend it to the first diagonal line, numbered 3, and proceed 
in the same manner for any other dimension. 

This is a very useful form of scale. The student should familiarize 
himself with its construction and application. 

Figure 5. 

Is AN INCH Diagonal Scale, divided into tenths and hundredths. 

It is made by drawing eleven lines parallel to each other, enclosing 
ten equal spaces, with vertical lines drawn through the points of 
division across the whole. The left hand vertical space is divided 
into ten equal parts, and diagonal lines drawn as in the figure. 

This scale gives three denominations. Each of the small spaces on 
the top and bottom lines, is equal to one-tenth of the whole divi- 
sion. The horizontal lines contained between the first diagonal 
and the vertical line, are divided into tenths of the smaller divi- 
sion, or hundreths of the larger division ; for example, the first 
line from the top contains nine-tenths of the smaller division, the 
second eight-tenths, the third seven-tenths, and so on as num- 
bered on the end of the scale. To make a dingonal scale of this 
form, divided into feet, inches, &:c., we must draw 13 paralli^l 
horizontal lines, and divide the left hand s|)ace also inl(^ 13. 



30 



PLATE X. 

CONSTRUCTION OF THE PROTRACTOR. 
Figure 1. 



The protractor is an instrument generally formed of a semicircle and 
its chord; the semicircle is divided into 180 equal parts or degrees^ 
numbered in both directions from 10° to 180°, as in its appHca- 
tion^ angles are often required to be measured or laid down on 
either hand; in portable cases of instruments the protractor is 
frequently drawn on a flat straight scale as in the diagram. Its 
mode of construction is sufficiently obvious from the drawing; a 
small notch or mark in the centre of the straight edge of the in- 
strument denotes the centre from which the semicircle is describ- 
ed^ and the angular point in which all the lines meet. 

Application. — Problem 33. Fig. 2. 



WITH THE PROTRACTOR, TO PROTRACT OR LAY DOWN 

ANY ANGLE. 

From the point let it he required to form a Right Angle to the 

line O. P. 

1st. Place the straight edge of the protractor to coincide with the 
line 0. P, with the centre at 0^ then mark the angle of 90° at S. 

2nd. From S draw S. 0^ which gives the required angle. 

While the instrument is in the position described^ with its centre 
at Oy any other angle may be laid down^ thus at Q we have 30°^ 
at R 60°, at T 120°^ and at V 150°^ and so on from the fraction 
of a degree up to 180.° 

The protractor may also be used for constructing any regular po- 
lygon in a circle or on a given line ; to do so, it is necessary to 
know the angle formed by said polygon by lines drawn from its 
corners to the centre of the circle, and also to know the angle 
formed by any two adjoining faces of the polygon. The table 
given for this purpose is constructed as follows : 

1st. To find the angle formed by any polygon at the centre, divide 
360, the number of degrees in the whole circle, by the number of 



Mate 10 
PKOTRACTOB. 

Its Construction and Application. 



Fic/. 1. 




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PLATE X. 



31 



sides in the required polygon^ the quotient will be the angle at 
the centre; for example^ let it be required to find the angle at the 
centre of an octagon: — divide 360 by 8^ the number of sides^ the 
quotient will be 45^ which is the angle formed by the octagon at 
the centre. 
2nd. To find the angle formed by two adjoining faces of a polygon^ 
we must subtract from 180 the number of degrees in the semi- 
circle^ the angle formed by said polygon at the centre^ the re- 
mainder will be the angle formed at the circumference. For ex- 
ample let us take the octagon ; we have found in the last paragraph 
that the angle formed by that figure in the centre is 45°; then if 
we subtract 45 from 180 it will leave 135^ which is the angle 
formed by two adjoining faces of the octagon. 



PLATE XL 



TO DESCRIBE FLAT SEGMENTS OF CIRCLES AND PARABOLAS. 



Very often in practice it would be very inconvenient to find the 
centre for describing a flat segment of a circle^ in consequence of 
the rise of an arch being so small compared to its span. 



Problem 34. Fig. 1. 



To describe a Segment with a Triangle, 

Note. — In all the diagrams in this plate A. B is the span of the arch, A. D 
the rise, and C the centre of the crown of the arch. 

1st. Make a triangle with its longest side equal to the chord or 
span of the arch and its height equal to one-half the rise. 

2nd. Stick a nail at A and C, place the triangle as in the diagram 
and move it round against the nails toward A. a pencil kept at the 
apex of the triangle will describe one-half of the curve. 

3rd. Stick another nail at B, and with the triangle moving against 
C and B, describe the other half of the curve. 



32 PLATE XI. 

Problem 35. Fig. 2. 



To describe the same Curve with strips of wood, forming a 

Triangle. 

] St. Drive a nail at Jl and another at B, place one strip against A 

and bring it up to the centre of the crown at C. 
2nd. Place another strip against B and crossing the first at C, nail 

them together at the intersection^ and nail a brace across to keep 

them in position. 
3rd. With the pencil at C and the triangle formed by the strips 

kept against ^ and B, describe the curve from C toward ^, and 

from C toward B. 



Problem 36. Fig. 3. 



To draw a Parabolic Curve by the intersection of lines forming 

Tangents to the Curve. 

1st. Draw C. 8 perpendicular to A. B, and make it equal to 
A. D. 

2nd. Join A. 8 and B. S, and divide both lines into the same 
number of equal parts^ say 8, number them as in the figure^ draw 
1. 1. — 2. 2. — 3. 3.^ &,c.^ then these lines will, be tangents to the 
curve ; trace the curve to touch the centre of each of those hnes 
between the points of intersection. 



Problem 37. Fig. 4. 



To draw the same Curve by another method. 

1st. Divide J.. D and B. E, into any number of equal parts^ and 
C. D and C. E into a similar number. 

2nd. Draw 1. 1. — 2. 2. &c.^ parallel to Jl. D, and from the points 
of division in A. D and B. E, draw lines to C. The points of 
intersection of the respective lines^ are points in the curve. 

Note. — The curves found, as in figs. 3 and 4, are quicker at the crown than 
a true circular segment ; but, where the rise of the arch is not more than 
one-tenth of the span, the variation cannot be perceived. 



Flatefl . 
FLAT SF.GME^^TS /JXD PARABOLAS. 




W ■■/t/.-.v/,- 



ricbte 12. 



OVAL FIGURES COMPOSED OF ARCS OF CIRCLES. 




Fig. 5. 






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PLATE XI. 



33 



Problem 38. Fig. 5, 



To describe a True Segment of a Circle by Intersections. 

1st Draw the chords ^. C and B, C, and J. and B. 0' ^ per- 
pendicular to them. 

2nd. Prolong D. E in each direction to 0. 0' ; divide 0. C^ C. 
0\ A. Dj A, 6j B. 6; and B. E into the same number of equal 
parts. 

3rd. Join the points 1. 1.— 2. 2. Slc, in J, B and 0. 0'. 

4th. From the divisions in J.. D, and B. E^ draw lines to C. 
The points of intersection of these lines with the former^ are 
points in the curve. A semicircle may be described by this method. 



PLATE XII. 

TO DESCRIBE OVAL FIGURES COMPOSED OF ARCS OF 

CIRCLES. 



Problem 39. Fig. 1. 



The length of the Oval A. B. being given, to describe an Oval 

upon it, 

1st. Divide A, B the given length, into three equal parts, in E 

and F, 
2nd. With one of those parts for a radius, and the compasses in 

E and F successively, describe two circles cutting each other in 

and 0^ 
3rd. From the points of intersection in and 0', draw lines 

through E and F, cutting the circles in V. V," and V.' J\"^ 
4th. With one foot of the compasses in 0, and 0' successively, and 

with a radius equal to 0. V," or OJ F, describe the arcs between 

V. V/ and V/' V", to complete the figure. 



Problem 40. Fig. 2. 



To describe the Oval^ the length A. B, and breadth C I), being 

given. 

1st. With half the breadth for a radius, and one foot in F, de- 
scribe the arc C. E, cutting .//. B in /:,'. 



34 



PLATE XII. 



2nd. Divide the difference E. B between the semiaxes into three 

equal parts^ and carry one of those divisions toward 4. 
3d. Take the distance B 4^ and set off on each side of the centre 

i^ at ^ and HJ 
4th. With the radius H, i7/ describe from H and H' as centres^ 

arcs cutting each other in K and KJ 
5th. From K and K,' through H and if/ draw indefinite right 

Hnes. 
6th. With the dividers in H^ and the radius H, A^ describe the 

curve V, Jl. V^' and with the dividers in H^ describe the curve 

VJ B. VJ" 
7th. From K and K^' with a radius equal to K. C, describe the 

curves V, C. V/ and VJ^ D, Vy" to complete the figure. 



Problem 41. Fig. 3. 



Another method for describing the Oval^ the length A. B^ and 

breadth C. D^ being given, 

1st. Draw C. B, and from B^ with half the transverse axiS; B, F, 
cut B. C m 0, 

2nd. Bisect B, by a perpendicular^ cutting A. B in P^ and C. 
D in Q. 

3rd. From Fj set off the distance F. P to R^ and the distance 
F. Q to S. 

4th. From S, through R and Py and from Q through R, draw 
indefinite lines. 

5th. From P and R, and from S and Q, describe the arcs^ com- 
pleting the figure as in the preceding problem. 

Note. — In all these diagrams, the result is nearly the same. Figs 1 and 2 
are similar figures, although each is produced by a different process. The 
proportions of an oval, drawn as figure 1, must always be the same as in 
the diagram ; but, in figs. 2 and 3, the proportions may be varied ; but, 
when the difference in the length of the axes, exceeds one-third of the longer 
one, the curves have a very unsightly appearance, as the change of curva- 
ture is too abrupt. These figures are often improperly called ellipses^ and 
sometimes false ellipses. Ovals are frequently used for bridges. When 
the arch is flat, the curve is described from more than two centres, but it is 
never so graceful as the true ellipsis. 



35 



PLATE XIII. 

TO DESCRIBE THE CYCLOID AND EPICYCLOID. 

The Cycloid is a curve formed by a point in the circumference of 
a circle, revolving on a level line ; this curve is described by any 
point in the wheel of a carriage when rolling on the ground. 

Problem 42. Fig. 1. 



To find any number of Points in the Cycloid Curve by the inter- 

section of lines, 

1st. Let G, H be the edge of a straight ruler, and C the centre 
of the generating circle. 

2nd. Through C draw the diameter A. B perpendicular to G, H, 
and E. F parallel to G. H; then A. B is the height of the curve, 
and E. F is the place of the centre of the generating circle at every 
point of its progress, 

3rd. Divide the semicircumference from B to A into any number 
of equal parts, say 8, and from Jl draw chords to the points of di- 
vision. 

4th. From C, with a space in the dividers equal to one of the di- 
visions on the circle, step off on each side the same number of 
spaces as the semicircumference is divided into, and through the 
points draw perpendiculars to G, H: number them as in the dia- 
gram. 

5th. From the points of division in E. F, with the radius of the 
generating circle, describe indefinite arcs as shewn by the dotted 
lines. 

6th. Take the chord A 1 in the dividers, and with the foot at 1 
and 1 on the line G. H, cut the indefinite arcs described from 1 
and 1 respectively at 1) and 1)^, then D and D' arc points in the 
curve. 

7th. With the chord J 2, from 2 and 2 in G. If, cut the indefinite 
arcs in ./ and J', with the chord JI 3, from 3 and 3, cut the arcs in 
K and A''' and apply the other chords in the same manner, cutting 
the arcs in L. JM, &lc. 

8th. Through the points so found trace the curve. 



36 PLATE XIII. 

Note. — The indefinite arcs in the diagram represent the circle at that point 
of its revolution, and the points D. /. K, &c., the position of- the genera- 
ting point B at each place. This curve is frequently used for the arches of 
bridges, its proportions are always constant, viz : the span is equal to the 
circumference of the generating circle and the rise equal to its diameter. 
Cycloidal arches are frequently constructed which are not true cycloids, but 
approach that curve in a greater or less degree. 



Figure 2. — The Epicycloid. 



This curve is formed by the revolution of a circle around a circle, 
either within or without its circumference, and described by a 
point B in the circumference of the revolving circle. P is the 
centre of the revolving circle, and Q of the stationary circle. 

Problem 43. 



To find Points in the Curve, 

1st. Draw the diameter 8. 8, and from Q the centre, draw Q: B 
at right angles to 8. 8. 

2nd. With the distance Q. P from Q, describe an arc 0. repre- 
senting the position of the centre P throughout its entire progress. 

3rd. Divide the semicircle B, D and the quadrants D. 8 into the 
same number of equal parts, draw chords from jD to 1, 2, 3, &.C., 
and from Q draw hues through the divisions in D, 8 to intersect 
the curve 0. in 1, 2, 3, &:c. 

4th. With the radius of P from 1, 2, 3, &,c., in 0. describe in- 
definite arcs, apply the chords D 1, Z) 2, &,c., from 1, 2, 3, Slc, 
in the circumference of Q, cutting the indefinite arcs in ^. C. E. 
jP, &c., which are points in the curve. 



PLATE XIV. 

DEFINITIONS OF SOLIDS. 

On referring back to our definitions, we find that a point has posi- 
tion without magnitude. 

A Line has length, without breadth or thickness, consequently 
has but one dimension. 



FlM&l^. 



CYCLOID AND EPICYCLOID. 



Fyl 





Flatcll. 
THE CUBE. 

its Section.'^- cuul I'^ir/nn' 



Fuf.l. 




') -<[-''l/ ■^■ 





Ficj. 3 




PLATE XIV. 



37 



A Surface has length and breadth^ without thickness^ conse- 
quently has two dimensions^ which^ multiplied together^ give the 
content of its surface. 

A Solid has lengthy breadth and thickness. These three dimen- 
sions multiphed together^ give its sohd content. 

Lineal Measure^ is the measure of lines. 

Superficial or Square Measure, the measure of surfaces. 

Cubic Measure, is the measure of sohds. 

For Example. — If we take a cube whose edge measures tw^o feet^ 
then two feet is the lineal dimension of that line. If the edge is 
two feet long^ the adjoining edge is also two feet long ; then, 
two feet multiplied by two, gives four feet, which is the superfi- 
cial content of a face of the cube. 

Then, if we multiply the square or superficial content, by two 
feet, which is the thickness of the cube, it will give eight feet, 
which is its solid content. 

Then, two lineal feet is the length of the edge. , 
^' four square '' the surface of one side. 

And eight cubic " the solid content of the cube. 



THE CUBE OR HEXAHEDRON, ITS SECTIONS AND SURFACE. 



Figure 1. 



1st. The cube is one of the regular polyhedrons, composed of six 
regular square faces, and bounded by twelve lines of equal 
length; the opposite sides are all parallel to each other. 

2nd. If a cube be cut through two of its opposite edges, and the 
diagonals of the faces connecting them, the section will be an 
oblong rectangular parallelogram, as fig. 2. 

3rd. If a cube be cut through the diagonals of three adjoining- 
faces, as in fig 3, the section will bean equilateral triangle, whose 
side is equal to the diagonal of a face of the cube. Two such 
sections may be made in a cube by cutting it again through the 
other three diagonals, and the second section will be ]iarallol to 
the first. 

4th. If a cube be cut by a plane passing through all its sides, the 
line of section, in each face, to be parallel with the diagonal, and 
midway between the diagonal and the corner of the face, as in 



3S PLATE XIV. 

fig. 4, the section will be a regular hexagon, and will be parallel 
with^ and exactly midway between the triangular sections de- 
fined in the last paragraph. 

5th. If a cube be cut by any other plane passing through all its 
sides^ the section will be an irregular hexagon. 

6th. The surface of the cube fig. 1^ is shewn at fig. 5, and if a 
piece of pasteboard be cut out, of that form^ and cut half through 
in the lines crossing the figure, then folded together, it will 
form the regular soHd. All the other solids may be made of 
pasteboard, in the same manner, if cut in the shape shewn in the 
coverings of the diagrams in the following plates. 

7th. The measure of the surface of a cube is six times the square 
of one of its sides. Thus, if the side of a cube be one foot, the 
surface of one side will be one square foot, and its whole surface 
would be six square feet. 

Its solidity would be one cubic foot. 

Note. — The cube may also, in general, be called a prisTUy and a parallelo- 
pipedon^ as it answers the description given of those bodies, but the terms 
are seldom applied to it. 



PLATE XV. 

SOLIDS AND THEIR COVERINGS 



Fig. 1. Is a sohd, bounded by six rectangular faces, each oppo- 
site pair being parallel, and equal to each other ; the sides are 
oblong parallelograms, and the ends are squares. It is called a 

right SQUARE PRISM, PARALLELOPIPED, Of PARALLELOPIPEDOX. 

j Fig. 2. Is its covering stretched out. 

Fig. 3. Is a triangular prism ; its sides are rectangles, and its 
ends equal triangles. 

Fig. 4. Is its covering. 

Prisms derive their names from the shape of their ends, and the 
angles of their faces, thus : Fig. 1 is a square prism, and fig. 2" 
a triangular prism. U the ends were pentagons, the prism 
would be pentagonal ; if the ends were hexagons, the prism 



Flatc 15 . 
SOLIDS AAW TEEIR COVEBIXGS . 



ly.l 



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Fui.n 




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Fig. 4. 







PLATE XV. 



39 



would be hexagonal^ Sfc. The sides of all regular prisms are 
equal rectangular parallelograms. I 

Fig. 5. Is a square pyramid^ bounded by a square at its 
base, and four regular triangles, as shewn at fig. 6. 

Pyramids, like prisms, derive their names from the shape of their 
bases ; thus we may have a square pyramid, as in fig. 4, or a ; 
triangular, pentagonal, or hexagonal pyramid, &c., as the base is 
a triangle, pentagon, hexagon, or any other figure. 

The sides of a pyramid incline together, forming a point at the 
top. This point is called its vertex^ apex^ or summit. 

The axis of a pyramid, is a line drawn from its summit, to the | 
centre of its base. The length of the axis, is the altitude of the i 
pyramid. When the base of a pyramid is perpendicular to its 
axis, it is called a right pyramid; if they are not perpendicular 
to each other, the pyramid is oblique. If the top of a pyramid be 
cut off, the lower portion is said to be truncated; it is also called 
a frustrum of a pyramid, and the upper portion is still a pyra- 
mid, although only a segment of the original pyramid. 

A pyramid may be divided into several truncated pyramids, or frus- 
trumSj and the upper portion remain a pyramid, as the name 
does not convey any idea of size, but a definite idea of form, 
viz : a solid, composed of an indefinite number of equal triangles, 
with their edges touching each other, forming a point at the top. 

A pyramid is said to be acute, right angled or obtuse, dependant 
on the form of its summit. 

An OBELISK is a pyramid whose height is very great compared 
to the breadth of the base. The top of an obelisk is generally 
truncated and cut off, so as to form a small pyramid, resting on 
the frustrum, which forms the lower part of the obelisk. 

When the polygon, forming the base of a pyramid, is irregular, 
the sides of the figure will be unequal, and the pyramid is called 
an irregular pyramid. 



40 



PLATE XVI. 

SOLIDS AND THEIR COVERINGS. 



Fig. 1. Is an hexago:n-al pyramid; and fig. 2 its covering. 

Fig. 3. A right cylinder, is bounded by two uniform circles, 
parallel to each other. The line connecting their centres, is 
called the axis. The sides of the cylinder is one uniform surface, 
connecting the circumferences of the circle, and everywhere 
equidistant from its axis. 



Problem 44. Fig. 4. 



To find the Le7igth of the Parallelogram A. B. C. D, to form the 

Side of the Cylinder. 

1st. Draw the ends, and divide one of them into any number of 

equal parts, say twelve. 
2nd. With the space of one of those parts, step off the same 

number on Jl. B, which will give the breadth of the covering 

to bend around the circles. 
Fig. 5. Is a right cojVe; its base is a circle, its sides sloping 

equally from the base to its summit. A Hne drawn from its 

summit to the centre of the base, is called its axis. If the axis 

and base are not perpendicular to each other, it forms an oblique, 

or scalene cone. 



Problem 45. Fig. 6. 



To draw the Coverings'. 



o 



1st. With a radius equal to the sloping height of the cone, from 

Ey describe an indefinite arc, and draw the radius E. F. 
2nd. Draw the circle of the base, and divide its circumference 

into any number of equal parts, say twelve. 
3rd. With one of those parts in the dividers, step off from F 

the same number of times to G, then draw the radius E. G, to 

complete the figure. 



FlatelS. 
SOLIDS AM) THE DEV ELOPEMENT OF THEIli SVJIEACES. 



Euj.L 



Fig. 2. 





ria. 





Us 'J Tz ' 



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W'Minin, 



Plate 17. 



SURFACES OF SOLIDS. 



FiQl. 




41 



PLATE XVII. 



COVERINGS OF SOLIDS 



Fig. 1. The Sphere 



Is a solid figure presenting a circular appearance when viewed in 
any direction; its surface is every where equidistant from a point 
within^ called its centre. 

1st, It may be formed by the revolution of a semicircle around its 
chord. 

2nd. The chord around which it revolves is called the axis,^ the 
ends of the axis are called poles. 

3rd. Any line passing through the centre of a sphere to opposite 
points, is called a diameter. 

4th. Every section of a sphere cut by a plane must be a circle, if 
the section pass through the centre^ its section will be a great cir- 
cle of the sphere ; any other section gives a lesser circle. 

5th. When a sphere is cut into two equal parts by a plane passing 
through its centre, each part is called a hemisphere ; any part of 
a sphere less than a hemisphere is called a segment ; this term 
may be applied to the larger portion as well as to the smaller. 



Problem 46. Fig. 2. 



To draw the Covering of the Sphere. 

1st. Divide the circumference into twelve equal parts. 

2nd. Step off on the line ^. B the same number of equal parts, 
and with a radius of nine of those parts, describe arcs through the 
points in each direction; these arcs will intersect each other in the 
lines C. D and E. F, and form the covering of the sphere. 



Figure 3. 



Is the surface of a regular Tetrahedkox, it is bounded by four 
equal equilateral triangles. 



42 plate xvii. 

Figure 4. 



The regular Octahedron is bounded by eight equal equilateral 



triangles. 



Figure 5. 



The Dodecahedron is bounded by twelve equal pentagons. 



Figure 6. 



The Icosahe{)Ron is bounded by twenty equal equilateral 

triangles. 
The four last figures^ together with the hexahedron delineated on 

Plate 14^ are all the regular polyhedrons. All the faces and all 

the solid angles of each figure are respectively equal. These solids 

are called platonic figures. 



PLATE XVIII. 

THE CYLINDER AND ITS SECTIONS. 



1st. If we suppose the right angled parallelogram A. B, C. D, 
fig. 1, to revolve around the side ^. B, it would describe a solid 
figure; the sides A. D and B, C w^ould describe two circles 
whose diameters would be equal to twice the length of the re- 
volving sides; the side C. D w^ould describe a uniform surface con- 
necting the opposite circles together throughout their whole cir- 
cumference. The solid so described would be a right cylinder. 

2nd. The line A. B, around which the parallelogram revolved^ is 
called the axis of the cylinder^ and as it connects the centres of 
the circles forming the ends of the cylinder^ it is every where 
equidistant from its sides. 

3rd. If the ends of a' cylinder be not at right angles to its axis^ it is 
called an oblique cylinder. 

4th. If a cylinder be cut by any plane parallel to its axis^ the sec- 
tion will be a parallelogram, as E. F. G, //, fig. 1. 



FlatelS 
TUT. CYZIXDER AXD SECTIOXS. 





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PLATE XVIII. 



43 



5th. If a cylinder be cut by any plane at right angles to its axis, 
the section will be a circle. 

6th. If a cylinder be cut by any plane not at right angles to its 
axis, passing through its opposite sides, as at K. L or M, JV^ fig. 
2, the section will be an ellipsis, of which the line of section 
K, L or M. JV would be the longest diameter, called the trans- 
verse or MAJOR diameter, and the diameter of the cylinder 
C, D would be the shortest diameter, called the conjugate or 

MINOR DIAMETER. 



Problem 47. Fig. 3. 



To describe an Ellipsis from the Cylinder with a string and 

two pins, 

1st. Draw the right lines JV. J\f and C. D at right angles to each 

other, cutting each other in S, 
2nd. Take in your dividers the distance P. M or P. JV, fig. 2, 

and set it off from S to M and JV*, fig. 3, which will make M, JV 

equal to M. JV, fig. 2. 
3d. From A. fig. 2, take A. D or A, C, and set it off from *Si to 

C and Z), making C. D equal to the diameter of the cylinder. 
4th. With a distance equal to S, M or S. JV from the points D 

and Cy cut the transverse diameter in E and F; then E and F 

are the foci for drawing the ellipsis. 
Note. — E is a focus, and P is a focus. E and F are foci. 
5th. In the foci, stick two pins, then pass a string around them, 

and tie the ends together at C. 
6th. Place the point of a pencil at C, and keeping the string 

tight, pass it around and describe the curve. 

Note. — The sum of all lines drawn from the foci, to any point in the curve, 
is always constant and equal to the major axis : thus, the length of the lines 
E. Rj and F. J?, added together, is equal to the lenjrth of E. C, and F. C, 
added together, or to two lines drawn from E and F, to any other point in 
the curve. 

7th. Fig. 4 is the section of the cylinder, through L. A', fig. 2, 
and is described in the same way as fig. 3. The letters of refer- 
ence are the same in both diagrams, except that the transverse 
diameter L. A", is made equal to the line of section L. h\ in 
fig. 2. 



44 PLATE XVIII. 

8th. The line JY. M, fig. 3, or L, K, fig. 4, is called the trans- 
verse^ or MAJOR Axis^ (plural axes^) and the line C. D, its 
CONJUGATE^ or MINOR AXIS. They are also called the transverse 
and conjugate diameters^ as above defined. The transverse axis 
is the longest line that can be drawn in an ellipsis. 

9th. Any line passing through the centre Sy of an ellipsis^ and 
meeting the curve at both extremities^ is called a diameter : 
every diameter divides the ellipsis into two equal parts. The 
CONJUGATE of any diameter, is a line drawn through the centre, 
terminated by the curve, parallel to a tangent of the curve at the 
vertex of the said diameter. The point where the diameter 
meets the curve, is the vertex of that diameter. 

10th. An ORDINATE to any diameter, is a line drawn parallel to 
its conjugate, and terminated by the curve and the said diameter. 
An ABSCISSA is that portion of a diameter intercepted between 
its vertex and ordinate. Unless otherwise expressed, ordinates 
are in general, referred to the axis, and taken as perpendicu- 
lar to it. Thus, in fig. 4, X. Y is the ordinate to, and L, X 
and K. X, the abscissae of the axis K. L.^ V. W is the ordinate 
to, and C. F, and D. F, the abscissae of the axis C, D, 



PLATE XIX. 

THE CONE AND ITS SECTIONS 
Definitions, 



1st. A CONE is a solid, generated by the revolution of a right 

angled triangle about one of its sides. 
2nd. If both legs of the triangle are equal, as S. JV* and JV*. 0, 

fig. 2, it would generate a right angled cone ; the angle S 

being a right angle. 
3rd. If the stationary side of the triangle be longest, as M, JV*, 

the cone will be acute, and if shortest, as T. JV, it will be 

OBTUSE angled. 
4th. The base of a cone is a circle, from which the sides slope 

regularly to a point, which is called its vertex, apex, or summit. 



I'blll' I. 'I 



nil-: coxi'i AM) ITS sj-jctio.ws. 




—'-^■'■- lin- , I lyiin" 



/■'/V/. 'A 



I'./ ■ 


/ I 


1 
1 


!• 


\ 


\K^ 






\ 




i". N I" i; I. 



I. .\ 



//""AA 



PLATE XIX. 



45 



5th. The axis of a cone, is a line passing from the vertex to the 
centre of the base, as M. JV^ figs. 1, 2, 3 and 4, and represents 
the hne about which the triangle is supposed to rotate. 

6th. A RIGHT CONE. When the axis of a cone is perpendicular 
to its base, it is called a right cone ; if they are not perpendicular 
to each other, it is called an oblique cone. 

7th. If a cone be cut by a plane passing through its vertex to the 
centre of its base, the section will be a triangle. 

8th. If cut by a plane, parallel to its base, the section will be a 
CIRCLE, as at E, F, fig. 1. 

9th. If the upper part of fig, 1 should be taken away, as at E. 
Fy the lower part would be a truncated cone, or frustrum, 
the part above E. F^ would still be a cone ; and, if another por- 
tion of the top were cut off from it, another truncated cone would 
be formed : thus a cone may be divided into several truncated 
cones, and the portion taken from the summit, would still remain 
a cone. Similar remarks have already been appUed to the 
pyramid. 

10th. If a cone be cut by any plane passing through its opposite 
sides, as at A, B, fig. 3, the section will be an ellipsis, 

11th. If a cone be cut by a plane, parallel with one of its sides, 
as at P, Q., R. S, or R, S'^ fig. 4, the section will be a para- 
bola. 

12th. If a cone be cut by a plane, which, if continued, would 
meet the opposite cone, as through C. D, fig. 4, meeting the op- 
posite cone at 0, the section will be an hyperbola. 



Problem 48. Fig. 3. 



To describe the Ellipsis from the Cone. 

1st. Let fig. 3 represent the elevation of a right cone, and .^. B 

the line of section. 
2nd. Bisect J. B in C, 
3rd. Through C, draw E. F perpendicular to the axis Af. jV, 

cutting the axis in P. 
4th. With one foot of the dividers in P, and a radius equal to 

P. E, or P. F, describe the arc E. D. F. 
5th. From C, the centre of the lino of section .//. B, draw C. D 

parallel to the axis, cutting the arc E. D. F in D, 



46 PLATE XIX. 

6th. Then A. B is the transverse axis, and C. D its semiconju- 
gate of an ellipsis^ which may be described with a string, as ex- 
plained for the section of the cyhnder, or by any of the other 
methods to be hereafter described. 

Note. — ^A section of the cylinder, as well as of a cone, passing through its 
opposite sides, is always an ellipsis. In the cone, the length of both axes 
vary with every section, but in the cylinder, the conjugate axis is always 
equal to the diameter of the cylinder, whatever may be the inclination of the 
line of section. 



Problem 49. Fig. 4. 



To find the length of the base line for describing the other sections, 

1st. With one foot of the dividers in JV^ and a radius equal to 
JY, T, or JV*. V^ describe a semicircle, equal to half the base of 
the cone. 

2nd. From C and P, the points where the sections intersect the 
base, draw P. A^ and C B, cutting the semicircle in A and B, 
Then A. P is one-half the base of the parabola, and C B is 
one-half the base of the hyperbola. The methods for describing 
these curves, are shewn in Plates 20 and 21. 



PLATE XX. 

TO DESCRIBE THE ELLIPSIS AND HYPERBOLA. 
Problem 50. Fig. 1. 



To find Points in the Curve of an Ellipsis by Intersecting Lines, 

Let A. By be the given transverse axis^ and C. D, the conjugate. 
1st. Describe the parallelogram L, M, JV. 0, the boundaries 

passing through the ends of the axes. 
2nd. Divide A, L, — A. JV, — B, M, and B, 0, into any number 

of equal parts^ say 4, and number them as in the diagram. 
3rd. Divide A, S, and B, S, also into 4 equal parts, and number 

them from the ends toward the centre. 



Plate 20 



/CLLIFSIS a:sd hyperbola 



PYc/. / 





W:"-Mws,fie-. 



Zirr.An&.Sons 



PLATE XX. 



47 



4th. From the divisions in A. L and B. M, draw lines to one 
end of the conjugate axis at C; and^ from the divisions in B. 
and ^, J\*y draw lines to the other end at D. 

5th. From D, through the points 1^2^ 3^ in Jl. S, draw lines to 
intersect the lines 1; 2, 3 drawn from the divisions on ^. L, and 
in hke manner through B, S, to intersect the hnes from B, M. 
These points of intersection are points in the curve. 

6th. From C, through the divisions 1^ 2^ 3^ on S, J. and S. B, 
draw lines to intersect the lines 1^ 2^ 3 drawn from Jl, JV and 
B, 0, which will give the points for drawing the other half of 
the curve. 

7th. Through the points of intersection^ trace the curve. 

Note. — If required on a small scale, the curve can be drawn by hand ; but, if 
required on a large scale, for practical purposes, it is best to drive sprigs at 
the points of intersection, and bend a thin flexible strip of pine around them, 
for the purpose of tracing the curve. Any number of points may be found 
by dividing the lines into the requisite number of parts. 



Figure 4. 



Is a semi-ellipsis^ drawn on the conjugate axis by the same method^ 

in which ^. B is the transverse^ and C, D the conjugate axis. 
Note. — This method will apply to an ellipsis of any length or breadth. 



Problem 51. Fig. 2. 



To draiv an Ellipsis with a Trammel. 

The TRAMMEL shcwn in the diagram is composed of two pieces 
of wood halfened together at right angles to each other, with a 
groove running through the centre of each, the groove being ] 
wider at the bottom than at the top. /. K. L is another strip o( 
wood with a point at I, or with a hole for inserting a pencil at /, 
and two sliding buttons at AT and L; the buttons are generally at-"] 
tached to small morticed blocks sliding over the strip, with wedges 
or screws for securing them in the proper jdacc; (the pins are 
only shewn in the diagiam,) the buttons attached to the pins are : 
made to slide freely in the grooves. 



48 PLATE XX. 

Mode of Setting the Trammel 

1st. Make the distance /. iiT equal to the semi-conjugate axis, and 
the distance from I to L equal to the semi-transverse axis. 

2nd. Set the grooved strips to coincide v^ith the axes of the ellipsis^ 
and secure them there. 

3rd. Move the point / around and it will trace the curve correctly. 

Note. — This is a very useful instmment, and was formerly used very fre- 
quently by carpenters to lay off their work, and also by plasterers to run their 
mouldings around elliptical arches, &c., the mould occupying the position 
of the point J. It was rare then to find a carpenter's shop without 
a trammel or to find a good workman who was not skilled in the use of it ; 
but since Grecian architecture with its horizontal lintels has taken the place 
of the arch, it is seldom a trammel is required, and when required, much 
more rare to find one to use ; but as it is sometimes wanted, and few of 
our young mechanics know how to apply it, at the risk of being thought 
tedious, we have been thus minute in its description. 



Problem 52. Fig. 3. 



To describe the Hyperbola from the Cone. 

1st. Draw the line Jl, C. B and make C B and C. A each equal 
to C. B, fig. 4, plate xix^ then A, B will be equal to the base of 
the hyperbola. 

2nd. Perpendicular to A, B, draw A. E and B. F, and make them 
equal to C. D, fig. 4, plate xix. 

3rd. Join E, F, from C erect a perpendicular C. D, 0^ and make 
C. equal to C. 0, fig. 4, plate xix. 

4th. Divide A. E and B. F each into any number of equal parts, 
say 4, and divide B, C and C. J. into the same number, and 
number them as in the diagram. 

5th. From the points of division on A. E and B. F, draw fines to D, 

6th. From the points of division in A. B, draw lines toward 0, and 
the points where they intersect the other lines with the same 
numbers will be points in the curve. The curve A. D, B is the 
section of the cone through the line C. D, fig. 4, plate xix. 



49 



PLATE XXI. 



PARABOLA AND ITS APPLICATION 



Problem 53. Fig. 1 



To describe the Parabola by Tangents, 

1st. Draw Ji, P. By make Jl, P and P. B each equal to ^, P, 

fig. 4y plate xix. 
2rid. From P draw P. Q. R perpendicular to Jl. B^ and make 

P. R equal to twice the height of P. Q, fig. 4^ plate xix. 
3rd. Draw A. R and B. R, and divide them each into the same 

number of equal parts, say eight; number one side from A to R, 

and the other side from R to B. 
4th. Join the points 1. 1. — 2. 2. — 3. 3, &.C.; the lines so drawn 

will be tangents to the curve, which should be traced to touch 

midway between the points of intersection. 
The curve A» Q. jB is a section of the cone through P. Q, fig. 4, 

plate xix. 



Problem 54, Fig. 2. 



To describe the Parabola by another method. 

Let ^x. J5 be the width of the base and P. Q the height of the 

curve. 

1st. Construct the parallelogram A. B. C. D. 
2nd. Divide Jl. Cand A. P — P. B and B. D respectively into a 

similar number of equal parts; number them as in the diagram. 
3rd. From the points of division in Jl. C and B. IK draw lines 

to Q. 
4th. From the ])oints of division on .//. B erect perpendiculars to 

intersect the other lines ; the points of intersection arc points in 

the curve. 



50 plate xxi. 

Problem 55, Fig. 3. 



To describe a Parabola by continued motion, with a Ruler y String 

and Square. 

Let C. D be the width of the curve and H. J the height. 

1st. Bisect H. D in K, draw J. K and K. E perpendicular to /. 

K, cutting /. H extended in E. Then take the distance H. E 

and set it off from J to Fy then F is the focus. 
2nd. At any convenient distance above J, fasten a ruler A, By 

parallel to the base of the parabola C. D. 
3rd. Place a square Sy with one side against the edge of the 

ruler^ J[. By the edge 0. JY of the square to coincide with the 

Hne E. J. 
4th. Tie one end of a string to a pin stuck in the focus at Fy place 

your pencil at Jy pass the string around it^ and bring it down to 

A'^y the end of the square^ and fasten it there. 
5th. With the pencil at Jy against the side of the square^ and the 

string kept tight^ slide the square along the edge of the ruler 

towards A; the pencil being kept against the edge of the square^ 

with the string stretched^ will describe one half of the parabola^ 

J, a 

6th. Turn the square over^ and draw the other half in the same 
manner. 



Definitions. 



1st. The FOCUS of a parabola is the point Fy about which the 

string revolves. The edge of the ruler A. By is the directrix of 

the parabola. 
2nd. The axis is the line /. Hy passing through the focus^ and 

perpendicular to the base C. D, 
3rd. The principal vertex^ is the point Jy where the top of the 

axis meets the curve. 
4th. The parameter^ is a hne passing through the focus^ parallel 

to the base^ terminated at each end by the curve. 
5th. Any line^ parallel to the axis^ and terminating in the curve^ 

is called a diameter^ and the point where it meets the curve^ is 

called the vertex of that diameter. 



Plate ZI. 



rAIL^BOLA. 





Fig. 3 



Fu/.L 




PLATE XXI. 



51 



Problem 56. Fig. 4. 



To apply the Parabola to the construction of Gothic Arches. 

1st. Draw Jl. B, and make it equal to the width of the arch 

at the base. 
2nd. Bisect A, B in E, draw E, F perpendicular to Jl. B^ and 

make E. F equal to the height of the arch. 
3rd. Construct the parallelogram A. B. C. D. 
4th. Divide E. F into any number of equal parts^ and D. F 

and F. C each into a similar number^ and number them as in 

the diagram. 
5th. From the divisions on F. D, draw lines to A, and from the 

divisions on F. Cy draw lines to B. 
6th. Through the divisions on E. F^ draw lines parallel to the 

base^ to intersect the other lines drawn from the same numbers 

on D. C. The points of intersection are points in the curve 

through which it may be traced. 

Note. — If we suppose this diagram to be cut through the line E. F, and 
turned around until E. A and E. B coincide, it will form a parabola, 
drawn by the same method as fig. 2 ; and, if we were to cut fig. 2 by tlie j 
line P. Q, and turn it around until P. A and P. B coincide, it would form ' 
a gothic arch, described by the same method as fig. 4 ; and, if the propor 
tions of the two figures were the same, the curves would exactly coincide. 



PLATE XXII. 



Problem 57. 



Given the position of three points in the circumference of a Cylin- 
der, and their respective heights from the hase^ to find the section 
of the segment of the Cylinder through these three points. 

1st. Let A. B. C be three points in the circumrerence of the base 
of the cylinder, immediately under the three given points, and 
A'. /)', — r. F, and B'. — E, — the height of the given points, 
respectively, above the base. 

2n(l. Join the points A and /?, and draw .//. A), — ('. /', ami />. 
E, perpendicuhir to A. B. 



52 



PLATE XXII. 



3rd. Make A. D equal to A^. U, the height of the given point 
above the base at A, — make B. ^ equal to J5^ E'j and C, F 
equal to C, P, 

4th. Produce B. Jl and E. D, to meet each other in H. 

5th. Draw C. G parallel to B, H, and F, G parallel to E. H. 

6th. Join G, H. 

7th. In G. H, take any point as G^ and draw G. K perpendicu- 
lar to G. C, cutting B. H in K, 

8th. From the point K^ draw K, I perpendicular to E. H, cut- 
ting E, H in L, 

9th. From Hy with the radius iJ. G^ describe the arc G. /, cut- 
ting K. L in /, and join H, I. 

10th. Divide the circumference of the segment A. C. B into 
any number of equal parts^ and from the points of division^ draw 
lines to A, B, parallel to G. H^ cutting A, B in l, 2^ 3^ &:c. 

11th. From the points 1; 2^ 3^ &c. in A B, draw Hnes parallel to 
B, E, cutting the line D, E in 1^ 2, 3^ &:c. 

12th. From the points 1^ 2^ 3, &c. in D. E^ draw lines parallel 
to H, I, and make 1. equal to 1. 1 on the base of the cyhn- 
der^ make 2. equal to 2. 2, 3. equal to 3. 3^ &c. 

13th. Through the points 0^ 0^ 0^ &:c.^ trace the curve^ which 
wnll be the contour of the section required. 

Note. — It will be perceived that the line 2. 2 intersects A. B in Jl, and that 
the line Jl. D obviates the necessity of drawing the perpendicular from 2, 
as required by the 1 1th step in the problem. 



PLATE XXIII. 



Problem 58. Fig 1. 



To draw the Boards for covering Circular Domes, 

To lay the boards vertically. Let A. D. C be half the plan 
of the dome ; let D. C represent one of the ribs^ and E. F the 
width of one of the boards. 

1st. Draw D, 0, and continue the line indefinitely toward H, 
2nd. Divide the rib D. C into any number of equal parts^ and 



FlrjXe 22 



TO FIND THE SECTION 



OF THE SEGMENT OF A CYLINDER 



THE O UGH THREE GIVEN POINTS. 




A ^ 



K. 



W"'Miiu/h 



riarr ?3 



TO l)R.4M THE BOARDS FOR COVERIXG 
HEMISPHERICAL DOMES. 




Fiij. :?. 




Tf^^'-mm* 



PLATE XXIII. 



53 



from the points of division^ draw lines parallel to A. C, meeting 
D. in 1, 2, 3, &c. 

3rd. With an opening of the dividers equal to one of the divi- 
sions on D. Cy step off from D toward H, the same number of 
parts as D. C is divided into^ making the right line D. H, nearly 
equal to the curve D, C. 

4th. Join E. and F. 0, 

5th. Make 1. c — 2. d — 3. e — 4./ and 5 g^ on each side of D. H^ 
equal to 1. c — 2. d — 3. e^ &c. on D, 0, 

6th. Through the points c. d, e. f. gy trace the curve, which will 
be an arc of a circle; and if a series of boards made in the same 
manner, be laid on the dome, the edges will coincide. 

Note. — In practice, where much accuracy is required, the rib should be di- 
vided into at least twelve parts. 



Problem 59. Fig. 2. 



To lay the hoards Horizontally, 

Let A. B. C be the vertical section of a dome through its axis. 

1st. Bisect A. C in D, and draw D. P perpendicular to A, C. 

2nd. Divide the arc A. B into such .a number of equal parts, that 
each division may be less than the breadth of a board. (If we 
suppose the boards to be used to be of a given length, each di- 
vision should be made so that the curves struck on the hollow 
side should touch the ends, and the curves on the convex side 
should touch the centre.) 

3rd. From the points of division, draw lines parallel to A. C to 
meet the opposite side of the section. Then if we suppose the 
curves intercepted by these Hues to be straight lines, (and the 
difference will be small,) each space would be the frustrum of a 
cone, whose vertex would be in the line D. P, and the vertex 
of each frustrum would be the centre from which to describe the 
curvature of the edges of the board to fit it. 

4th. From 1 draw a line through the point 2, to meet the line /) 
P in E; then from Ey with a radius equal to E. 1, describe the 
curve 1. L, which will give the lower edge of the board, an J 
with a radius equal to E. 2, describe the arc 2. A", which will 
give the upper edge. The line L. K drawn to E^ will give the cut 
for the end of a board which will fit the end of any other board 
cut to the same angle. 



54 



PLATE XXIII. 



0th. From 2 draw a line through 3^ meeting D. P in F. From 
3^ draw a line through 4^ meeting D. P in G^ and proceed for 
each board^ as in paragraph 4. 

6th. If from C we draw a hne through M^ and continue it up- 
ward^ it would require to be draw^n a very great distance before 
it would meet D. P ; the centre w^ould consequently be incon- 
veniently distant. 

For the bottom board^ proceed as follows : 

1st. Join A. M^ cutting D. P in JY, and join JST, 1. 

2nd. Describe a curve^ by the methods in Problems 34 or 35^ 
Plate 11; through 1. JV. M^ which will give the centre of the 
board^from which the width on either side may be traced. 



PLATE XXIV. 

CONSTRUCTION OF ARCHES 



Arches in architecture are composed of a number of stones arra.ng- 
ed symmetrically over an opening intended for a door^ window^ 
&.C.; for the purpose of supporting a superincumbent weight. The 
depth of the stones are made to vary to suit each particular case^ 
being made deeper in proportion as the width of an opening be- 
comes larger^ or as the load to be supported is increased; the size 
of the stones also depends mxuch on the quality of the material of 
which they are composed : if formed of soft sandstone they will 
require to be much deeper than if formed of granite or some other 
hard strong stone. 

Defixitiojvs. Fig. 2. 



1st. The SPAN of an arch is the distance between the points of 
support; which is generally the width of the opening to be cov- 
ered; as A. B. These points are called the springing points ; the 
mass against which the arch rests is called the abutmejs't. 

3rd. The risE; height or versed siive of an arch; is the dis- 
tance from C to D. 
\ 2nd. The springing line of an arch is the line Jl. B, being a 
horizontal line drawn across the tops of the support where the 
arch commences. 



Plate .24 



JOINTS IN ARCHES 



/'Yr/.l 



/'Yr/. 2 




\ J-: ]•" G 



Fig. J. 



Fig. 4. 



K 




PLATE XXIV. 



55 



4th. The crown of an arch is the highest pointy as D. 
5th. VoussoiRS is the name given to the stones forming the arch. 
6th. The keystone is the centre or uppermost voussoir D^ so 
called^ because it is the last stone set^ and wedges or keys the 
wliole together. Keystones are frequently allowed to project 
from the face of the wall^ and in some buildings are very elabor- 
ately sculptured. 
7th. The intrados or soffit of an arch is the under side of 
the voussoirs forming the curve. 
8th. The extrados or back is the upper side of the voussoirs. 
9th. The thrust of an arch is the tendency which all arches have 
to descend in the middle; and to overturn or thrust asunder the 
points of support. 

Note. — The amount of the thrust of an arch depends on the proportions be- 
tween the rise and the span, that is to say, the span and weight to be sup- 
ported being definite; the thrust will be diminished in proportion as the 
rise of the arch is increased, and the thrust will be increased in proportion 
as the crown of the arch is lowered. 

10th. The JOINTS of an arch are the lines formed by the adjoining 
faces of the voussoirs ; these should generally radiate to some de- 
finite point; and each should be perpendicular to a tangent to the 
curve at each joint. In all curves composed of arcs of circles, a 
tangent to the curve at any point would be perpendicular to a 
radius drawn from the centre of the circle through that point, 
consequently the joints in all such arches should radiate to the 
centre of the circle of which the curve forms a part. 

11th. The BED of an arch is the top of the abutment; the shape 
of the bed depends on the quality of the curve, and will be ex- 
plained in the diagrams. 

12th. A RAMPANT ARCH is ouc in which the springing points are 
not in the same level. 

13th. A STRAIGHT ARCH, or as it is more properly called, a plat 
BAND, is formed of a row of wedge-shaped stones of equal depth 
placed in a horizontal line, the upper ends of the stones being 
broader than the lower, prevents them from foiling into the void 
below. 

Hth. Arches are named from the shape of the curve of the under 
side, and are either simple or complex. I would define simple 
curves to be those that are struck from one centre, as any segment 
of a circle, or by continued motion, as the ellipsis, parabola, hy- 
perbola, cycloid and epicycloid; and complex arches to be 



^^ PLATE XXIV. 

those described from two or more fixed centres^ as many of the 
Gothic or pointed arches. The simple curves have all been de- 
scribed in our problems of practical Geometry; we shall however 
repeat some of them for the purpose of showing the method of 
drawing the joints. 



Problem 60. Fig. 1. 



To describe a Segment or Scheme Jlrch, and to draw the Joints, 

1st. Let E and F be the abutments^ and the centre for describ- 
ing the curve. 

2nd. With one foot of the dividers in 0, and the distance 0. F^ 
describe the line of the intrados. 

3rd. Set off the depth of the voussoirs^ and with the dividers at 
Oy describe the line of the extrados. 

4th. From E and F draw lines radiating to 0, w^hich gives the 
line of the bed of the arch. This Hne is often called by masons 
a skew-back. 

5th. Divide the intrados or extrados^ into as many parts as you 
design to have stones in the arch^ and radiate all the Hues to 0, 
which will give the proper direction of the joints, 

6th. If the point should be at too great a distance to strike the 
curve conveniently, it may be struck by Problem 34 or 35, 
Plate 1 1 ; and the joints may be found as follows : Let it be de- 
sired to draw a joint at 2, on the line of the extrados ; from 2 
set off any distance on either side, as at 1 and 3; and from 1 and 
3, with any radius, draw two arcs intersecting each other at 4 — 
then from 4 through 2 draw the joint which will be perpendicular 
to a tangent, touching the curve at 2. This process must be re- 
peated for each joint. The keystone projects a little above and 
below the lines of the arch. 



Prob. 6L Fig. 2. — The Semicircular Arch. 



This requires but little explanation. J[. B is the span and C the 
centre, from which the curves are struck, and to which the lines 
of all the joints radiate. The centre C being in the springing 
line of the arch the beds of the arch are horizontal. 



PLATE XXIV. 



67 



Prob. 62. Fig. 3. — The Horse Shoe Arch 



Is an arc of a circle greater than a semicircle^ the centre being 
above the springing line. 

This arch is also called the Saracenic or Moresco arch^ because 
of its frequent use in these styles of architecture. The joints 
radiate to the centre^ as in fig. 2. The joint at 5^ below the 
horizontal line^ also radiates to 0. This may do very well for a 
mere ornamental arch, that has no weight to sustain; but if, as in 
the diagram, the first stone rests on a horizontal bed, it would be 
larger on the inside than on the outside, and would be liable to 
be forced out of its position by a shght pressure, much more so 
than if the joint were made horizontal, as at 6. These remarks 
will also apply to fig. 4, Plate 25. 



Problem 63. Fig. 4. 



To describe an Ogee Jlrch^ or an Jlrch of Contrary Flexure. 

Note. — This arch is seldom used over a large opening, but occurs frequently 
in canopies and tracery in Gothic architecture, the rib of the arch being 
moulded. 

1st. Let A. B be the outside width of the arch, and C. D the 

height, and let A. E be the breadth of the rib. 
2nd. Bisect A. B in C, and erect the perpendicular C. D ; bisect 

A. C in F, and draw F. J parallel to C. D. 
3rd. Through D draw J. K parallel to A. B, and make D. K 

equal to D. J. 
4th. From F set off F. G, equal to A, E the breadth of the rib, 

and make C. H equal to C. G. 
5th. Join G. J and H. K; then G and H will be the centres for 

drawing the lower portion of the arch, and J and K will be the 

centres for describing the upper portion, and the contrary curves 

will meet in the lines G. J and H. K. 



Problem 64. Fig. 5. 



To draw the Joints in an Elliptic Arch: 

Let A. B be the span of the arch, C. D the rise, and F. F the 
foci, from which the line of the intrados may be described. 



58 PLATE XXIV. 

The voussoirs near the spring of the arch are increased in depth^ 
as they have to bear more strain than those nearer the crown ; 
the outer curve is also an elUpsis^ of which i/and i/are the foci. 

To draw a joint in any part of the curve^ say at 5. 

1st. From F and F the foci^ draw hnes cutting each other in the 
given point 5^ and continue them out indefinitely. 

2nd. Bisect the angle 5 by Problem 11^ Page 18^ the line of 
bisection will be the line of the joint. 

The joints are found at the points 1 and 3 in the same manner. 

3d. If we bisect the internal angle^ as for the joints 2 and 4^ the 
result will be the same. 

4th. To draw the corresponding joints on the opposite side of 
the arch^ proceed as follows : 

5th. Prolong the line C, D indefinitely toward E, and prolong 
the lines of bisection 1^ 2^ 3^ 4 and 5^ to intersect C. E in 1^ 2^ 
3^ &.c.^ and from those points draw the corresponding joints be- 
tween A and D, 



PLATE XXV. 

TO DESCRIBE GOTHIC ARCHES AND TO DRAW THE JOINTS. 



The most simple form of Pointed or Gothic arches are those com- 
posed of two arcs of circles^ whose centres are in the springing 
line. 



Figure 1. — The Lancet Arch. 



When the length of the span A. B is much less than the length 
of the chord A, C, as in the diagram^ the centres for striking the 
curves will be some distance beyond the base^ as shewn by the 
rods; the joints all radiate to the opposite centres. 

. Fig. 2. — The Equilateral Arch. 



When the span D. E, and the chords D. F and E. F form an 
equilateral triangle^ the arch is said to be equilateral^ and the 



Flate 2 -J 



FOTNIEJJ AliCIIES 



Fuj.l. 



Fifj. 2 




/■?(/. J 



/')(/. / 




PLATE XXV. 



59 



centres are the points D and E in the base of the arch^ to which 
all the joints radiate. 

Fig. 3. — The Depressed Arch 



Has its centres within the base of the arch^ the chords being shorter 
than the span ; the joints radiate to the centres respectively. 

Note. — There are no definite proportions for Gothic arches, except for the 
equilateral ; they vary from the most acute to those whose centres nearly 
touch, and which deviate but little from a semicircle. 

Fig. 4. — The Pointed Horseshoe Arch. 

This diagram requires no explanation ; the centres are above the 
springing line. See fig. 3^ plate xxiv^ page ^7, 

Figure 5. 



To describe the Four Centred Pointed Arch. 

1st. Let A. B be the springing line^ and E. C the height of the 
arch. 

2nd. Draw B. D parallel to E. C, and make it equal to two- 
thirds of the height of E. C. 

3rd. Join D. C, and from C draw C. L perpendicular to C. D. 

4th. Make C. G and B, F both equal to B. D. 

5th. Join G. F^ and bisect it in H^ then through H draw //. L 
perpendicular to G, F meeting C, L in L. 

6th. Join L. F, and continue the line to JV. Then L and F are 
the centres for describing one-half of the arch, and the curves will 
meet in the line L. F. JY. 

7th. Draw L. M parallel to A. B, make 0. ,M equal to 0. L, 
and E. K equal to E. F, Then A^ and M are the centres for 
completing the arch, and the curves will meet in the line M. K. P, 

8th. The joints from P to C will radiate to Al; from C to *'\'tlu\v 
will radiate to L; from JV to B they will railinte to /'\ and from 
P to ^ they will radiate to A''. 

Note 1. — As the joint at P radiativs to both the rcutros K and .W, and ihr 
joint at JV radiates holli to /''and /., \hr chaiiL;*' of direction o\ \\\v \o\\cr 
joints is easy and pleasinp^ to tlio eye, so niuidi so that wr shoidd he \incon- 
scious of the change, if the construct ivi* lines wcyc riMnovcd. 



60 



PLATE XXV. 



Note 2. — When the centres for striking the two centred arch are in the 
springing line, as in diagrams 1, 2 and 3, the vertical side of the opening 
joins the curve, without forming an unpleasant angle, as it would do if the 
vertical lines were continued up above the line of the centres ; it is true that 
examples of this character may be cited in Gothic buildings, but its ungrace- 
ful appearance should lead us to avoid it. 



PLATES XXVI AND XXVII 



DESIGN FOR A COTTAGE. 



Fig. 1 . Is the elevation of the south-east front. 

Fig. 2. Plan of the ground floor. 

Fig. 3. Section through the line E, F on the plan fig. 2^ the 
front part of the house supposed to be taken away. 

Fig. 4. Plan of the chamber floor. 

This simple design is given for the purpose of shewing the method 
of drawing ihe plans ^ section, elevation and details of a building; 
it is not offered as a ^^ model cottage/^ although it would make a 
very comfortable residence for a small family. 

The PLAN of a building is a horizontal section; if we suppose the 
house cut off* just above the sills of the windows of the second 
floor and the upper portion taken away^ it would expose to view 
the whole interior arrangement^ shewing the thickness of the 
walls, the situation and thickness of the partitions and the position 
of doors^ windows^ &.C.; all these interior arrangements are in- 
tended to be represented by fig. 4. 

If we perform the same operation above the sills of the first floor 
windows, the arrangements of that floor^ including the stairs and 
piazzas, would appear as in fig. 2. 

A SECTION of a building is a vertical plan in which the thicknesses 
of the walls^ sections of the fire-places and flues, sizes and direc- 
tion of the timbers for the floors and roof, depth of the foundations 
and heights of the stories are shewn, all drawn to a uniform scale. 

If the front of the building is supposed to be removed, as in fig. 3, 
the whole of the inside of the back wall will be seen in elevation, 
shewing the size and finish of the doors and windows, the height 
of the washboards, and the stucco cornice in the parlor. In looking 



PLATES XXVI AND XXVII. 



61 



through the door at K, the first flight of stairs in the back build- 
ing is seen in elevation. 

If we suppose the spectator to be looking in the opposite direction^ 
the back part of the house removed^ he would see the inside of 
the front windows &c. instead of the back. 

An ELEVATION of a building is a drawing of the front^ side or back^ 
in which every part is laid down to a scale^ and from which the 
size of every object may be measured. 

A PERSPECTIVE VIEW of a buildiug^ is a drawing representing it 
as it would appear to a spectator from some definite point of view^ 
and in which^ all objects are diminished as they recede from the 
eye. 

The PLANS^ SECTIONS and elevations^ give the true size and 
arrangements of the building drawn to a scale^ and shew the 
whole construction. 

The perspective view should shew the building complete, in 
connection with the surrounding objects^ which would enable the 
proprietor to judge of the effect of his intended improvement. 

The whole constitutes the design^ which for a country house can- 
not be considered complete without a perspective view. 

To make a design for a dwelling house^ or other building^ it is ne- 
cessary before we commence the drawing^ that we should know 
the site on which it is to be erected^ and the amount of accommo- 
dation required. 

In choosing a site for a country residence many subjects should be 
taken into consideration; for example^ it should be easy of access^ 
have a good supply of pure water^ be on elevated ground to allow 
the rain and other water to flow freely from it^ but not so high as 
to be exposed to the full blasts of the winter storms; it should 
have a good prospect of the surrounding country^ and above all, 
it should be in a salubrious locality, free from the malaria arising 
from the vicinity of low or marshy grounds, with free scope to 
allow the house to front toward the most eligible point of the 
compass. i 

The ASPECT of a country house is of much importance; for if tlu^ i 
site commands an extensive view, or pleasant prospect in any di- 
rection, the windows of the sitting and principal sleeping rooms, 
should front in that direction : provided it does not also face the 
point from which blow the prevailing storms of the climate, this 
should be particularly considered in choosing the site. Rooms to 
be cheerful and pleasant, should front south of due east or west ; at 



62 PLATES XXVI AND XXVII. 

the same time it is desirable that the view of disagreeable or un- 
sightly objects should be excluded^ and as many as possible of the 
agreeable and beautiful objects of the neighborhood brought into 
view. The design before us^ is made to front the south-east^ all 
the openings except two are excluded from the north easterly 
storms, which are the most disagreeable in the Atlantic States ; 
the sun at noon would be opposite the angle A, and w^ould shine 
equally on the front and side, consequently the front would have 
the sun until the middle of the afternoon, and the side of the front 
house and front of the back building w^ould have the evening sun, 
rendering the whole dry and pleasant. 

The end of the back building, containing the kitchen and stairs, is 
placed against the middle of the back wall of the front building, 
to allow the back windows in the parlor, &c., to be placed in the 
middle of each room. These windows may be closed in stormy 
weather with substantial shutters; but in warm weather they will 
add much to the coolness of the rooms, by allowing a thorough 
ventilation. 

The broad projecting cornice of the house, and the continuous 
piazza, are the most important features in the elevation; besides 
the advantage of keeping the walls dry, and throwing the rain- 
water away from the foundation, they give an air of comfort, 
which would be entirely wanting without them ; for if w^e w^ere 
to take away the piazza, and reduce the eave cornice to a slight 
projection, the appearance would be bald and meagre in the 
extreme. 

The projection of the piazza is increased on the front and rear, to 
give more room to the entrances. 

The front building is 36 feet wide from A to B^ fig. 2, and 20 feet 
deep from A to D. The back building is 16 feet wide, by 20 
feet deep. The scale at the bottom of each plate must be used 
to get the sizes of all the minor parts. The height of the first 
story is 10 feet in the clear, and of the second story 8^ Q"] these 
heights are laid off on a rod i?, to the right of fig. 1 ; so are also 
the heights of the windows, which sheAvs at a glance, their posi- 
tion with regard to the floors and ceilings. This method should 
always be resorted to in drawing an elevation, as it will the better 
enable the draughtsman to make room for the interior finish of 
the windows and for the cornice of the room. 

In laying down a plan, the whole of the outer walls should be first 
drawn, and in setting off openings and party walls, the measure- 



PLATES XXVI AND XXVII. 



63 



merits should be taken from both corners, to prove that you are 
correct. For example, in setting off the front door, take the 
width 5. from the whole width of the front, w^hich will leave 
31. 0; then lay off 15^ Q" from Jl, and also from By then if the 
width of 5. is left between the points so measured, you are 
sure the front door is laid off correctly ; as the windows C and H 
are midway between the front door and the corner of the build- 
ing, the same plan should be followed, and as a general rule 
that will save trouble by preventing errors, you should never de- 
pend on the measurements from one end or corner ^ if you have 
the means of proving them by measuring from the opposite end 
also. 

The winding steps in the stairs may be dispensed with by adding 
3 steps to the bottom flight bringing it out to the kitchen door, 
and by adding 1 step to the top flight ; or a still better arrange- 
ment might be made by adding 3 steps to the bottom flight, and 
retaining two of the winders: this would give 17 risers instead of 
16, the present number, w^hich would reduce the height of each 
to 7 3-4 inches. 

To ascertain the number of steps required to a story, proceed as 
follows : Add to the clear height of the story the breadth of the 
joists and floor, w^hich will give the full height from the top of 
one floor to the top of the next. In constructing the stairs this 
height is laid off" on a rod, and then divided into the requisite 
number of risers; but in drawing the plan, as in the case before 
us, set down the height in feet, inches and parts, and divide by 
the height you propose for your rise: this will give you the 
number of risers. If there is any remainder, it may be divided 
and added to your proposed rise, or another step may be added, 
and the height of the rise reduced ; or the height of the story 
may be divided by the number of risers, which will give the exact 
height of the riser in inches and parts. For example : 

The clear height of the story in the design is 10^0.'^ ^ , ,, ^„ 

The breadth of the joist and thickness of the floor \. 0. S 
this multiplied by 12 would give 132 inches, and 132 inches di- 
vided by 16, the number of risers on fig. 1, will give 8 1-4 inches; 
or divided by 17 would give 7 3-4 inches and a fraction. As the 
floor of the upper story forms one step, there will be always one 
tread less than there arc risers. The vertical front of each step 
is called the rise or riscr^ and the horizontal part is called the 
tread or step. When the eaves of the house are continued 



64 PLATES XXVI AND XXVII. 

around the building in the same horizontal line as in this design, 
the roof is said to be hipped^ and the rafter running from the cor- 
ner of the roof diagonally to the ridge is called the hip rafter. 



REFERENCES TO THE DRAWINGS. 



Similar letters in the plans and sections refer to the same parts : thus 
T the fire-place of the parlor in fig. 2, is shewn in section at T^ 
fig. 3; and M the plan of the back parlor window in fig. 2, is 
shewn in elevation at M^ fig. 3. 

A. By fig. 2, is the plan, and A. B, fig. 1, the elevation of the 
front wall. 

E. F, fig. 2, the line of the section. 

Gj fig. 2y the front door. 

K, fig. 2, the plan, and K, fig. 3, the elevation of the door leading 
to the back building. 

Z, fig. 2j the plan, and Z, fig. 3, the elevation of the first flight of 
stairs. 

M and JV, fig. 2, the plans, and M and JV, fig. 3, elevation of the 
back first floor windows. 

and P, fig. 4, the plans, and and P, fig. 3, elevation of cham- 
ber windows. 

Q, fig. 2, the plan, and Q, fig. 3, section of the parlor side window. 

P, fig. 4, the plan, and R, fig. 3, section of chamber side window. 

>S, fig. 4, the plan, and S^ fig. 3, elevation of railing on the landing. 

Ty fig. 2, the plan, and J", fig. 3, section of parlor fire-place. 

Uy fig. 2, the plan, and Z7, fig. 3, section of breakfast room fire-place. 

V and Wy fig. 4, the plans, and V and Wy fig. 3, sections of cham- 
ber fire-place. 

X and Yy fig. 2, the plans, and X and F, fig. 3, elevations of side 
posts on piazza. 

Z. Z. Zy fig. Ay plans of closets. 

a. a. a.y fig. 4, flues from fire-places of ground floor. 

h. by fig. 3, section of eave cornice. 

c. c, fig. 3, rafters of building. 

d. dy fig. 3, rafters of piazza. 

e. e. e. e. e. e. e. e. e, joists of the different stories; the ends of the 
short joists framed around the fire-places and flues are shewn in 
dark sections ; the projection around the outside walls of fig. 4, 
shews the roof of the piazza. 



Flate ^6. 



DESIGN EOH A COTTAGE 



r 1 



Pu 




A 



A 



I'll! I I I I I I I I I- 



Yca/c of Tret 



.30 



I'TIC-Mzmn 



I^lati^ ;?7 



IJJ^SfGN FOR J rOTTAGE 




Hate \'d 



DETAILS OF COTTAGE 



7r(27is\ erse Section 
oflioofajid Cormct 




-I 
J 



T- 



It- 





Fla7i of Joists for First Floor 



j-u/. :^ 



65 



PLATE XXVIII. 



Details. 

Fig. 1 is an elevation of one pair of rafters^ shewing also a section 
through the cornice and top of the wall. 
Ay section of the top of the wall. 

By ceiling joist^ the outside end notched to receive the cornice. 
C, collar beam. D. D, rafters. 
E, raising plate. F, wall plate. 
G, cantilever and section of cornice. 



Figure 2. 



Plan of First Floor Joists. 

A J foundation of kitchen chimney. 

By foundation of parlor chimney; Cy of breakfast room do. 

Dy double joist to receive the partition dividing the stairway from 

the kitchen. 
E. Ey &.C. double joists resting on the walls and supporting the 

short joists F, F. Fy forming the framing around the fire-places. 
The joists E. E. E and Dy are called trimming joists. 
The short joists F. F. F are called triramerSy and the joists G. G. 

Gy framed into the trimmers with one end resting on the wall are 

called tail joists. 



PLATE XXIX. 



Details. 

Fig. 1, horizontal section through the parlor window. 
Jly is the outside of the wall. B the inside of wall. 



66 PLATE XXIX. 

Cy the hanging stile of sash frame. 

D^ the inside hning. E the outside Hning. 

F^ the back hning. G, G the weights. 

Hy the stile* of the outside or top sash. 

/; the stile of the inside or bottom sash. 

K^ inside stop bead. L^ jamb lining. 

My ground. f JY^ plastering. 

Oj architrave. Fy (dotted hne) the projection of the plinth. 

* The stiles of a sash, door, or any other piece of framing, are the vertical 

outside pieces; the horizontal pieces are called rails, 
f Grounds are strips of wood nailed against the wall to regulate the thickness 

of the plastering, and to receive the casings or plinth. 



Figure 2. 



Vertical Section through the Sills. 

Ay outside of the wall. 

Qy stone sill of the window. 

Ry wooden sub-sill. 

>S^^ bottom rail of sash. Ty bondtimber 

Uy framing under window^ called the back. 

Vy cap of the back. Ky the inside stop bead. 



Figure 3. 



Flinth of Parlor, 

M, My grounds. JVy plastering. 

Vy plinth or washboard. Wy the base moulding. 

Xy the floor. 

Many more detail drawings might be made of this design^ and 
where a contract is to be entered into^ many more should be made. 
Enough is here given to explain the method of drawing them; 
their use is to shew the construction of each part^ and when 
drawn to a large scale, as in plate xxix, a workman of any in- 
telligence would be able to get out any part of the work required. 



Flat£ 2.9. 



DETAILS OF COTTAGE 

One fourth of full size. 



Horizontal section through Parlor windrnv. 

ya. 1. 







K 




i -^^^^ 



JM:i»;;.-x:w, ;yi/'^,;%^ 



r 



\frU(o/ Sri/noi 
l/iniin//( f/ir .Si//.v 



1^ 






■^xx#^'s^&-?.^ 



M 



~L 



M 



hii.r, 



W'^M: 



Flate 30 



OCTAGONAL PLAN AND ELEVATION. 




i 




EUhrAlioJi: I'h/. '^ 




Flate 3J 



CIRCULAR FLAN AND ELEVATION 




% 



67 



PLATE XXX. 



OCTAGONAL PLAN AND ELEVATION. 



Fig. 1. — Half the Plan. Fig. 2. — Elevation. 



This plate requires but little explanation^ as the dotted lines from 
the different points of the plan^ perfectly elucidate the mode of 



drawing the elevation. 



The dotted line ^^ shews the direction of the rays of hght by 
which the shadows are projected ; the mode of their projection 
will be explained in Plates 55 and 56. 



PLATE XXXI. 



CIRCULAR PLAN AND ELEVATION 



This plate shews the mode of putting circular objects in elevation. 
The dotted lines from the different points of the plan, determine 
the widths of the jambs (sides) of the door and windows, and 
the projections of the sills and cornice. One window is farther 
from the door than the other, for the purpose of shewing the 
different apparent widths of openings, as they are more or less 
inclined from the front of the picture. 

This, as well as Plate 30, should be drawn to a much larger size 
by the learner; he should also vary the position and width of the 
openings. As these designs are not intended for a particular 
purpose, any scale of equal parts may be used in drawing them. 



68 



PLATE XXXII. 

ROMAN MOULDINGS 



Roman mouldings are composed of straight lines and arcs of 
circles. 

Note. — Each separate part of a moulding, and each moulding in an assem- 
blage of mouldings, is called a member. 



Fig. 1. — A Fillet^ Band or Listel 



Is a raised square member^ with its face parallel to the surface on 
which it is placed. 

Fig. 2. — Bead. 



A moulding whose surface is a semicircle struck from the centre K. 

« 

Fig. 3.— Torus. 



Composed of a semicircle and a fillet. The projection of the 
circle beyond the fillet^ is equal to the radius of the circle which 
is shewn by the dotted line passing through the centre L. The 
curved dotted line above the fillet, and the square dotted fine be- 
low the circle, shew the position of those members when used 
as the base of a Doric column. 



Fig. 4. — The Scotia 



Is composed of two quadrants of circles between two fillets. B 
is the centre -for describing the large quadrant ; A the centre for 
describing the small quadrant. The upper portion may be made 
larger or smaller than in the diagram, but the centre A must 
always be in the line B, A, The scotia is rarely, if ever used 
alone ; but it forms an important member in the bases of columns. 



Plate 32. 



ROMAX yiOULDIXGS. 



FiUet. 



Fig.l. 




Bead 



7'u/.?. ^ 



Fig. 3. Torn. 



Fig. 4. Scotia 





Fig. J. 



OvoU 



Ficj. 6. ' Co.vctto 




Fig. 7. Cvnia Becta 



Fi(j. 8. (I'm a Bevei^sa 



\% 





\\ 



PLATE XXXII. 



69 



Fig. 5. — The Ovolo 



Is composed of a quadrant between two fillets. C is the centre for 
describing the quadrant. The upper fillet projects beyond the 
curvC; and by its broad shadow adds much to the effect of 
the moulding. The ovolo is generally used as a bed moulding, 
or in some other position where it supports another member. 



Fig 6. — The Cavetto, 



Like the ovolo, is composed of a quadrant and two fillets. The 
concave quadrant is used for the cavetto described from D ; it 
is consequently the reverse of the ovolo. The cavetto is frequently 
used in connection with the ovolo, from which it is separated by 
a fillet. It is also used sometimes as a crown moulding of a cor- 
nice; the crown moulding is the uppermost member. 

Fig. 7. — The Cyma Recta 



Is composed of two arcs of circles forming a waved line, and 
two fillets. 

To describe the cyma, let / be the upper fillet and JV the lower 
fillet. 

1st. Bisect /. JY, in M, 

2nd. With the radius M. JV or M. /, and the foot of the divi- 
ders in JY and M^ successively describe two arcs cutting each 
other in jP, and from M and / with the same radius, describe 
two arcs, cutting each other in E, 

3rd. With the same radius from E and F^ describe two arcs 
meeting each other in M. 

The proportions of this moulding may be varied at pleasure, by 
varying the projection of the upper fillet. 



Fig. 8. — The Cyma Reversa, Talon or Ogee. 



Like the cyma recta, it is composed of two circular arcs and two 
fillets; the upper fillet projects beyond the curve, and the lower 
fillet recedes within it. 

The curves are described from G and //. 

The CYMA, or cyma recta has the concave curve ii])pormost. 



70 PLATE XXXII. 

The CYMA REVERSA has the concave curve below. 

The CYMA RECTA is used as the upper member of an assemblage 

of mouldings^ for which it is well fitted from its light appearance. 
The CYMA REVERSA from its strong form^ is like the ovolo^ used to 

sustain other members. 
The dotted lines drawn at an angle of 45° to each moulding, 

shew the direction of the rays of light, from which the shadows 

are projected. 
Note. — When the surface of a moulding is carved or sculptured, it is said 

to be ENRICHED. 



PLATE XXXIII. 



GRECIAN MOULDINGS 

Are composed of some of the curves formed by the sections of a 
cone, and are said to be elliptic, parabolic, or hyperbohc, taking 
their names from the curves of which they are formed. 



Figures 1 and 2. 



To draw the Grecian Echinus or Ovolo, the fillets A and B, the 
tangent C. B, and the point of greatest projection at D being 
given, 

1st. Draw B, H, a continuation of the upper edge of the under 

fillet. 
2nd. Through D, draw D, H perpendicular to B, Hy cutting the 

tangent B, C in C 
3rd. Through B, draw B. G parallel to D. H, and through 2), 

draw D. E parallel to H. B, cutting G. B in E, 
4th. Make E. G equal to E, B, and E. F equal to H, C, join 

D. F. 
5th. Divide the lines D. jPand D. C each into the same number of 

equal parts. 
6th. From the point B, draw lines to the divisions 1, 2, 3, &c. 

in D, a 



Plate .'/.'J. 



aiU'iCLAX M o r a ijtxgs . 



E'c7zi?iiLs- or Ovolo 




Fig.l. 




I'ui. ?. 



Cy/?ia Hccla 




lu/. ./. 



3 Z 1 1 



/vy. /. 




F ' - R 



i I'll If/ //V'/vv.s// 



^rotiti 





.V / 



/'if/, ft 



It'-'M' 



PLATE XXXIII. 



71 



7th. From the point G, draw lines through the divisions in D, F^ 

to intersect the hnes drawn from B, 
8th. Through the points of intersection trace the curve. 

Note. — A great variety of form may be given to the echinus, by varying the 
projections, and the incHnation of the tangent B. C. 

Note 2. — If H. C is less than C. D, as in fig. 1, the curve will be elliptic; 
if H. C and C. D are equal, as in fig. 2, the curve is parabolic ; if H, C be 
made greater than D. C, the curve will be hyperbolic. 

Note 3. — The echinus, when enriched with carving, is generally cut into 
figures resembling eggs, with a dart or tongue between them. 



Figs. 3 and 4. — The Grecian Cyma. 



To describe the Cyma Becta, the perpendicular height B. D and 
the projection A. D being given. 

1st. Draw Jl. C and B. D perpendicular to ^. D and C. B par- 
allel to A. D. 

2nd. Bisect J. D in E, and J. C in G; draw E. F and G. 0, 
which w^ill divide the rectangle Jl. C. B. D into four equal 
rec tangles. 

3rd. Make G. P and 0. K each equal to 0. H. 

4th. Divide Jl. G — 0. B — Ji. £'and B. i^into a similar number 
of equal parts. 

5th. From the divisions in A. E and F. B, draw lines to H; 
from P draw lines through the divisions on A. G to intersect the 
lines drawn from A. E, and from K through the divisions in 0. 
By draw lines to intersect the lines drawn from F. B. 

6th. Through the points of intersection draw the curve. 

Note. — The curve is formed of two equal converse arcs of an ellipsis, of 
which E. Fis the transverse axis, and P. H or H. K the conjugate. The 
points in the curve are found in the same manner as in fig. 1, plate 20. 

Fig. 5. — The Grecian Cyma Reversa^ Talon or Ogee. 



To draw the Cyma Rcvcrsa^ the fillet A^ the point C, the end of 
the curve B, and the line B. D being gircn. 

1st. From Cy draw C. D, and from 7?, draw B. E perpendicular 
to B. D, then draw C. E parallel to B. 1), which completes the 
rectangle. 



72 PLATE XXXIII. 

2nd. Divide the rectangle B, E, C. D into four equal partS; by 

drawing F. G and 0. P. 
3rd. Find the points in the curve as in figs. 3 and 4. 

Note 1. — If we turn the figure over so as to bring the line F. G vertical, G 
being at the top, the point B of fig. 5, to coincide with the point A of fig. 3, 
it will be perceived that the curves are similar, F. G. being the transverse 
axis, and JY. H or M. H the conjugate axis of the ellipsis. 

Note 2. — The nearer the line B. D approaches to a horizontal position, the 
greater will be the degree of curvature, the conjugate axis of the ellipsis 
will be lengthened, and the curve become more like the Roman ogee. 

Figure 6. — The Grecian Scotia. 



To describe the Grecian Scotia, the position of the fillets A and 

B being given. 

1st. Join Jl, ^^ bisect it in C^ and through C drsiw D, jE^ parallel 

to B, G, 
2nd. Make C. D and C. E each equal to the depth intended to be 

given to the scotia; then Jl. B v^ill be a diameter of an ellipsis, 

and D. E its conjugate. 
3rd. Through E, draw F. G parallel to Jl. B. 
4th. Divide A. F and B. G into the same number of equal parts, 

and from the points of division draw lines to E. 
5th. Divide A. C and B. C into the same number of equal parts, 

as A. F, then from D through the points of division in A. J5, draw 

lines to intersect the others, which will give points in the curve. 



PLATE XXXIV. 

PLAN, SECTION AND ELEVATION OF A WHEEL AND PINION. 

The cross Hnes on Q. i?, fig. 2, shewing the teeth of the wheel 
and pinion, are drawn from the elevation as described in Plate 31, 
which explains the method of drawing an elevation from a circu- 
lar plan. 

This plate is introduced to give the learner an example for draw- 
ing machinery ; it requires but little explanation, as the relative 



Plate 34 



ELEVATION. 




Fig. 3 Section. 




I,\r^'^-Mir7_m&. 



lUrnan kSons 



PLATE XXXIV. 



73 



parts are plain and simple; the same letters refer to the same 

parts in each figure. 
Thus J. J fig. 1^ is the end of the axle of the wheel. 
Jl. B, fig. 2, the top of the axle of the wheel. 
^. B, fig. 3^ section through the centre of the wheel. 
C. D, the axle of the pinion. 

E, F, flanges of the barrel^ with the rope coiled between them. 
G. H, bottom piece of frame. 
/. K. K, JYy inclined uprights of frame. 
Ly top of frame. M. M, top cross pieces of frame. 
0, P, bearings of the wheel. 
Q. R, plan and elevation of wheel. 
Rj intersection of wheel and pinion. 
S, S, bottom cross pieces of frame. 
When two wheels engage each other^ the smallest is called a pinion. 



PLATE XXXV. 



TO DRAW THE TEETH OF WHEELS 



1st. The LijfE of CENTRES is the Hne ^. B, D, fig. I^ passing 
through K and C, the centres of the wheel and pinion. 

2nd. The proportiojn'al or primitive diameter of the wheel^ is 
the line Jl. B ; the proportional radius A. K or K. B. The true 
radii are the distances from the centres to the extremities of the 
teeth. 

3rd. The proportional diameter of the pinion is the Hne B. 
D ; the proportional radius C. B. 

4th. The proportional circles or pitch lines are circles de- 
scribed with the proportional radii touching each other in B. 

5th. The pitch of a wheel is the distance on the pitch circle in- 
cluding a tooth and a space, as E. F or G. IL or (). D^ fig. 2. 

6th. The depth of a tooth is the distance from the pitch circle to 
the bottom, as L. K, fig. 1, and the height of a tooth is the dis- 
tance from the pitch circle to the top of the tooth, as L. ./)/. 



10 



74 



PLATE XXXV. 



To draw the Pitch Line of a Pinion to contain a definite number 
of Teeth of the same size as in the given wheel K, 



1st. Divide the proportional diameter A, B of the given wheel 

into as many equal parts as the wheel has teeth^ viz. 16. 
2nd. With a distance equal to one of these parts^ step oiF on the 

line B, D as many steps as the pinion is to contain teeth^ which 

will give the proportional diameter of the pinion ; the diagram 

contains 8. 
3rd. Draw the pitch circle^ and on it with the distance E, Fy the 

pitchy lay off the teeth. 
4th. Sub-divide the pitch for the tooth and space, draw the sides 

of the teeth below the pitch line toward the centre, and on the 

tops of the teeth describe epicycloids. 

Note'. — The circumferences of circles are directly as their diameters; if the 
diameter of one circle be four times greater than another the circumference 
will also be four times greater. 

Fig. 2 is another method for drawing the teeth; A, B is the pitch 
circle on which the width of the teeth and spaces must be laid 
down. Then with a radius D. E or D. F^ equal to a pitch and 
a fourth, from the middle of each tooth on the pitch circle as at D, 
describe the tops of the teeth E and Fy from describe the tops 
of the teeth G and D, and so on for the others. The sides of 
the teeth within the pitch circle may be drawn toward the centre, 
as at F and if, or from the centre 0, with a radius equal to 0. 
Q or 0. P, describe the lower part of the teeth G and D, 




Plate 35. 



TEETH OE WHEELS 




y 




W" Mr 



Flate36 



CYLIXDER OF A LOCOMOTBE 

Scale '8 ^^ of full size 





1 



:x 



I 



>• I 



/vy./ 




Tup I'f'ii' 




L 


A 


_, 










i 





^-a»8»»»»»;»«aa»;«»»^a8aa8«^-«»»a;»<i<»»gi<»'^s»<»>»««a»8«»>»^-^»^^ 



jy^/. y. 



Section tlirniiqh J.lj. FiqJ. 



Pla/c .,'7 



CYLmPEB OF A LOCOMOTTVE 

Scale /s^^.^ of /hi/ size 





F/q.J. End view 




////. /.\r///(>// f/iionif/i //.'// /'/// / /'/(//e.l/i 



75 



PLATES XXXVI AND XXXVII. 



PLAN, SECTIONS AND END ELEVATION OF A CYLINDER FOR 

A LOCOMOTIVE ENGINE. 



Fig. 1. Top view or Plan. 

Fig. 2. Longitudinal Section through Jl, By fig. 1, 

Fig. 3. Elevation of the end B^ fig. 1. 

Fig. 4. Transverse Section through G. H^ fig. 1. 



REFERENCES. 



A. 

A, J5. 
C. 

K 

F. 

G.H. 

H. 

K, 

X. 

M.M, 

JV. 



Stuffing box. 



Line of longitudinal section. 



Steam exhaust port^ or Exhaust. 

Steam ports or Side openings. 

Piston rod. 

Piston shev^^n in elevation. 

Line of transverse section. 

Exhaust pipe. 

Packing. 

Gland or Follower. 

Heads of cylinder. 

Valve face. 

The piston is represented in the drawing as descending to the bot- 
tom of the cylinder; the bent arrows from D to C, fig. 2, and 
from C to Hj fig. 4, shew the course of the steam escaping from 
the cylinder through the steam port and exhaust port to the ex- 
haust pipe ; the other arrow at D, (\g. 2, the direction of the 
steam entering the cylinder. 



76 



PLATE XXXVIII. 

ISOMETRICAL DRAWING 
Figure 1. 



To draiv the Isometrical Cube- 

Let A be the centre of the proposed drawing. 

1st. With one foot of the dividers in ^, and any radius, describe a 
circle. 

2nd. Through the centre ^^ draw a diameter B, C parallel to the 
sides of the paper. 

3rd. With the radius from the points B and C lay off the other 
corners of a hexagon, D. E. F. G, 

4th. Join the points and complete the hexagon 

5th. From the centre A^ draw lines to the alternate corners of the 
hexagon, which will complete the figure. 

The isometrical cube is a hexahedron supposed to be viewed at an 
infinite distance, and in the direction of the diagonal of the cube ; 
in the diagram, the eye is supposed to be placed opposite the 
point A: if a wire be run through the point A to the opposite 
corner of the cube, the eye being in the same line, could only see 
the end of the wire, and this would be the case no matter how 
large the cube, consequently the front top corner of the cube and 
the bottom back corner must be represented by a dot, as at the 
point A. As the cube is a sohd, the eye from that direction will 
see three of its sides and nine of its twelve edges, and as the dis- 
tance is infinite, all these edges will be of equal length, the edges 
seen are those shewn in fig- 1 by continuous lines; three of the 
edges and three of the sides could not be seen, these edges are 
shewn by dotted lines in fig. 1, but if the cube were transparent 
all the edges and sides could be seen. The apparent opposite 
angles in each side are equal, two of them being 120°, and the 
other two 60° ; all the opposite boundary lines are parallel to each 
other, and as they are all of equal length may be measured by 
one common scaky and all fines parallel to any of the edges of 
the cube may be measured by the same scale. The lines F. G, 
A. C and E. D represent the vertical edges of the cube, the par- 



/'/r//r :/fj 



COXSTRVCTIOX OF I'llE /SOMETIUCAL CL'BE 



J' If/. /. 




Plate 39. 



ISOJIETRICAL FIGIHES 



Fuj.l. 



Fig. 2. 






W^Mimfir. 



Jlimankoom 



PLATE XXXVIII. 



77 



allelograms A, C. D. E and A. C, F. G, represent the vertical 
faces of the cube^ and th-e parallelogram A. B. E. F represents 
the horizontal face of the cube; consequently vertical as well as 
horizontal lines and surfaces may be delineated by this method 
and measured by the same scale^ for this reason the term isome- 
TRicAL (equally measurable) has been apphed to this style of 
drawing. 

Figure 2 



Is a cube of the same size as fig. 1, shaded to make the represen- 
tation more obvious; the sides of the small cube A^ and the 
boundary of the square platform on which the cube rests^ as well 
as of the joists which support the floor of the platform^ are all 
drawn parallel to some of the edges of the cube^ and forms a good 
illustration for the learner to practice on a larger scale. 

Note. — A singular optical illusion may be witnessed while looking at this 
diagram, if we keep the eye fixed on the point Ay and imagine the drawing 
to represent the interior of a room, the point v3 will appear to recede; then 
if we again imagine it to be a cube the point will appear to advance, and 
this rising and falling may be continued, as you imagine the angle A to rep- 
resent a projecting corner, or an internal angle. 



PLATE XXXIX. 

EXAMPLES IN ISOMETRICAL DRAWING. 



Figs. 1 and 2 are plans of cubes with portions cut away. Figs. 
3 and 4 are isometrical representations of them. 

To draw a part of a regular figure^ as in these diagrams, it is bet- 
ter to draw the whole outline in pencil, as shewn by the dotted 
lines, and from the corners lay off* the indentations. 

The circumscribing cube may be drawn as in ^v^, 1, Plate 38, 
with a radius equal to the side of the plan, or with a triangle 
having one right angle, one angle of 60°, and the other angle 
30°, as shewn at A, Proceed as follows : — 

Let B be the tongue of a square >or a straight edge apj^hod hori- 
zontally across the paper, apply the hy{)othenuse of tlie triangle 
to the tongue or straight edge, as in the dingrani, and draw the 



78 ' ' PLATE XL. 

left hand inclined lines ; then reverse the triangle and draw the 
right hand inclined lines ; turn the short side of the triangle 
against the tongue of the square^ and the vertical hnes may be 
drawn. 
This instrument so simplifies isometrical drawing^ that its applica- 
tion is but little more difficult than the drawing of flat geometrical 
plans or elevations. 



PLATE XL. 

EXAMPLES IN ISOMETRICAL DRAWING—CONTINUED. 



Fig 1 is the side^ and fig, 2 the end elevation of a block pierced 
through as shewn in fig 1^ and with the top chamfered off, as 
shewn in figs. 1 and 2. 

Figure 3. 



To draw the figure Isometrically. 

1st. Draw the isometrical lines ^. B and C D ; make A. B 

equal to A. ^ fig. 1; and C. D equal to C D fig. 2. 
2nd. From Jl, B and D^ draw the vertical lines, and make them 

equal to J5. G^ fig. 1. 
3rd. Draw K. H and L, I parallel to Jl, B^ and if. I and K, L 

parallel to (7. Z). 
4th. Draw the diagonals H. D and /. C^ and through their inter- 
section draw a vertical line M, G. F, Make G. F equal to G. 

F, fig. 1. 
5th. Through G, draw G. JV*, intersecting L, K in JV^ and from 

JV' draw a vertical line JV. E, 
6th. Through F, draw F. E, intersecting JY. E in E; then E. F 

represents the fine E. F in fig. 1 . 
7th. From E and Fy lay off the distances and P^ and from 

and P draw the edges of the chamfer 0. K — 0. L — P. J? and 

P. I, which complete the oudine. 
8th. On ^. B lay off the opening shewn in fig, 1, and from R^ 

draw a line parallel to C, D. 



Plate 40 



ISOMETRIC4L FIGURES 



K — 



Fu/. J. 



V, 11, 



Fill. ? 




Fiq. 3. 



^~A I 



i'.i. I'i'.i'i' ■! 

mm 




r. ( 



/'/>/. ^. 




W" Mmi/ie. 



PLATE XL. 



79 



Note 1. — All the lines in this figure, except the diagonals and edges of the 
chamfer, can be drawn with the triangle and square, as explained in 
Plate 39. 

Note 2. — All these lines may be measured by the same scale, except the in- 
clined edges of the chamfer, which will require a different scale. 

Note 3. — The intelligent student will easily perceive from this figure, how 
to draw a house with a hipped roof, placing the doors, windows, &c., each 
in its proper place ; or how to draw any other rectangular figure. In- 
clined lines may always be found by a similar process to that we have pur- 
sued in drawing the edges of the chamfer. 

Figure 4 



Is the elevation of the side of a cube with a large portion cut out. 



Figure 5 



Is the isometrical drawing of the same^ with the top of the cube 
also pierced through. The mode pursued is so obvious^ that it 
requires no explanation : it is given as an illustration for drawing 
FURNITURE^ or any other framed object. It requires but little 
ingenuity to convert fig. 5 into the frame of a table or a foot-stool. 



PLATE XLI. 



TO DRAW THE ISOMETRICAL CIRCLE 



Figure 1 



Is the plan of a circle inscribed in a square^ with two diameters 
•^. B and C D parallel to the sides of the square. 



Figure 2. 



To draw the Isometrical Representation. 

1st. Draw the isometrical square, M. JV, 0. 1\ having its opposite 
angles 120^ and 60° respectively. 
2nd. Bisect each side and draw JI. B and C. D, 



80 



PLATE XLI. 



3rd. From draw 0. J and 0. D, and from M draw M, C and 

J^. B intersecting in Q and R, 
4th. From Q^ with the radius Q. *^^ describe the arc A, C^ and 

from R^ with the same radius^ describe the arc D, B. 
5th. From 0, with the radius 0. ^^ draw the arc ^. D^ and from 

Mj with the same radius^ describe C. B, which completes the oval. 

Note. — ^An isometrical projection of a circle would be an ellipsis ; but 
the figure produced by the above method is so simple in its construction and 
approaches so near to an ellipsis, that it may be used in most cases, besides 
its facility of construction, its circumference is so nearly equal to the cir- 
cumference of the given circle, that any divisions traced on the one may be 
transferred to the other with sufficient accuracy for all practical purposes. 

Figure 3. 



To divide the Circumference of the Isometrical Circle into any 

number of equal parts. 

1st. Draw the circle and a square around it as in fig. 2^ the square 
may touch the circle as in fig. 2^ or be drawn outside as in fig. 3. 

2nd. From the middle of one of the sides as 0^ erect 0. K per- 
pendicular to E. Fy and make 0. K equal to 0. E, 

3rd. Draw K. E and K. F^ and from K with any radius^ describe 
an arc P. Qy cutting K. E in Py and K. F in Q. 

4th. Divide the arc P 4 into one-eighth of the number of parts re- 
quired in the whole circumference^ and from K^ through these di- 
visions^ draw fines intersecting E. in 1, 2 and 3. 

5th. From the divisions 1^ 2 and 3^ in E, 0, draw fines to the 
centre P^ which wifi divide the arc E. into four equal parts. 

6th. Transfer the divisions on E. from the corners E, F, G. H, 
and draw lines to the centre P, when the concentric curves will 
be divided into 32 equal parts. 

Note 1. — If a plan of a circle divided into any number of equal parts be 
drawn, as that of a cog wheel, the same measures may be transferred to the 
isometric curve as explained in the note to fig. 2, but if the plan be not 
drawn, the divisions can be made as in fig. 3. 

Note 2. — The term isometrical projection has been avoided, as the pro- 
jection of a figure would require a smaller scale to be used than the scale to 
which the geometrical plans and elevations are drawn, but as the isometri- 
cal figure drawn with the same scale to which the plans are drawn, is in 
every respect proportional to the true projection, and conveys to the eye the 
sam^ view of the object, it is manifestly much more convenient for practical 
purposes to draw both to the same scale. 



riate 41 



ISOMETRICAL CIRCLE 




IV'" Mnii.fi f . 



PLATE XLI. 



81 



Note 3. — In Note 2 to fig. 3, Plate 40, allusion has been made to inclined 
lines requiring a different scale from any of the lines used in drawing the 
isometric cube: for the mode of drawing those scales as well as for the further 
prosecution of this branch of drawing, the student is referred to Jopling's 
and Sopwith's treatise on the subject, as we only propose to give an intro- 
duction to isometrical drawing. Sufficient, however, has been given to en- 
able the student to apply it to a very large class of objects, and it w^ould ex- 
tend the size of this work too much (already much larger than was intend- 
ed) if we pursue the subject in full. 



PERSPECTIVE. 



PLATE XLII. 



The design of the art of perspective is to draw on a plane surface 
the representation of an object or objects, so that the representa- 
tion shall convey to the eye^the same image as the objects them- 
selves would do if placed in the same relative position. 

To elucidate this definition it will be necessary to explain the mail- 
ner in which the image of external objects is conveyed to the eye. 

1st. To enable a person to see any object, it is necessary that such 
object should reflect light. 

2nd. Light reflected from a centre becomes weaker in a duplicate 
ratio of distance from its source, it being only one-fourth as in-' 
tense at double the distance, and one-ninth at triple the distance, 
and soon. 

3rd. A ray of light striking on any plane surface, is reflected from 

.that surface in exactly the same angle with which it impinges; 
thus if a plane surface be placed at an angle of 45°, to the direc- 
tion of rays of light, the rays will be reflected at an angle of 45"^ 
in the opposite direction. This fact is expressed as follows, viz : 

THE ANGLK OF REFLECTION IS EQUAL TO THE ANGLE OF 

n 



82 PLATE XLII. 

INCIDENCE. This axiom, so short and pithy, should be stored 
in the memory with some others that we propose to give, to be 
brought forward and apphed whenever required. 

4th. Rays of light reflected from a body proceed in straight lines 
until interrupted by meeting with other bodies, which by reflec- 
tion or refraction, change their direction. 

5th. Refraction of light. When a ray of hght passes from a 
rare to a more dense medium, as from a clear atmosphere through 
a fog or from the air into water, it is bent out of its direct course : 
thus if w^e thrust a rod into w^ater, it appears broken or bent at the 
surface of the water ; objects have been seen through a fog by the 
bending of the rays, that could not possibly be seen in clear wea- 
ther; this bending of the rays' of Hght is called refraction, and the 
rays are said to be refracted : this effect, (produced however by a 
different cause) may often be seen by looking through common 
window^ glass, when in consequence of the irregularities of its 
surface, the view^ of objects without is much distorted. 

6th. A portion of light is absorbed by all bodies receiving it on 
their surface, consequently the amount of light reflected from an 
object is not equal to the quantity received. 

7th. The amount of absorption is not the same in all bodies, but 
depends on the color and quality of the reflecting surface ; if a ray 
falls on the bright polished surface of a looking-glass, most of it 
will be reflected, but if it should fall on a surface of black cloth, 
most of it would be absorbed. White or hght colors reflect more 
of a given ray of light than dark colors ; polished surfaces reflect 
more than those which are unpolished, and smooth surfaces more 
than rough. 

8th. As all objects absorb more or less light, it follows that at each 
reflection the ray will become weaker until it is no longer per- 
ceptible. 

9th. Rays received from a luminous source are called direct, and 
the parts of an object receiving these direct rays are said to be in 
LIGHT. The portions of the surface so situated as not to receive 
the direct rays are said to be in shade ; if the object receiving 
the direct rays is opaque, it will prevent the rays from passing in 
that direction, and the outline of its illuminated parts will be pro- 
jected on the nearest adjoining surface : the figure so projected is 
called its shadow. 

10th. The parts of an object in shade will always be lighter than 
the shadow, as the object receives more or less reflected light from 



PLATE XLII. 



83 



the atmosphere and adjoining objects, the quantity depending on 
the position of the shaded surface, and on the position and quali- 
ty of the surrounding objects. 
I 11th. If an object were so situated as to receive only a direct ray 
of light, without receiving reflected light from other sources, the 
illuminated portion could alone be seen ; but for this universal law 
of reflection we should be able to see nothing that is not illumi- 
nated by the direct rays of the sun or by some artificial means, and 
all beyond would be one gloomy blank. 

12th. Rays of light proceeding in straight lines from the surfaces 
of objects, meet in the front of the eye of the spectator where 
they cross each other, and form an inverted image on the back of 
the eye of all objects within the scope of vision. 

13th. The size of the image so formed on the retina depends on 
the size and distance of the original ; the shape of the image de- 
pends on the angle at which it is seen. 

Note. — The size of objects diminishes directly as the distance increases, ap- 
pearing at ten times the distance, only a tenth part as large; the knowledge 
of this fact has produced a system oi arithmetical perspective, which enables 
the draughtsman to proportion the sizes of objects by calculation. 

14th. The strength of the image depends on the degree of illumi- 
nation of the original, and on its distance from the eye, objects 
becoming more dim as they recede from the spectator. 

15th. To give a better idea of the operation of the eye in viewing 
an object, let us refer to fig. 1. The circle A is intended to repre- 
sent a section of the human eye, H the pupil in front, K the 
crystalline lens in which the rays are all converged and cross each 
other, and M the concave surface of the back of the eye called 
the retina^ on which the image is projected. 

16th. Let us suppose the eye to be viewing the cross B, C, and 
that the parallelogram JV, 0, P, Q represents a picture frame 
in which a pane of glass is inserted; the surface of the glass slight- 
ly obscured so as to allow objects to be traced on it, then rays 
from every part of the cross will proceed in straight lines to the 
eye, and form the inverted image C. B on the retina. If with a 
pencil we were to trace the form of the cross on the glass so as to 
interrupt the view of the original object, we should have a true 
perspective representation of the original, which would form ex- 
actly the same sized image on the retina; thus the point h would 
intercept the view of B^ c of C, rf of D and e of E^ and if colored 
the same as the original, the image formed from it would be the 
same in every respect as from the original. 



84 



PLATE XLII. 



17th. If we move the cross B. C to F. G, the image formed on 
the retina would be much larger^ as shewn at G. F^ and the rep- 
resentation on the glass would be larger^ the ray from F passing 
through/^ and the ray from G passing through g^ shewing that 
the same object will produce a larger or smaller image on the re- 
tina as it advances to or recedes from the spectator; the farther it 
recedes^ the smaller will be the image formed^ until it becomes so 
small as to be invisible. 

18th. Fig. 2 is given to elucidate the same subject. If we suppose a 
person to be seated in a room^ the ground outside to be on a level 
with the bottom of the window Jl. B^ the eye at >S' in the same 
level line^ and a series of rods C. D, E. F of the same height 
of the window to be planted outside^ the window to be filled 
with four lights of glass of equal size^ then the ray from the bot- 
tom of all the rods would pass through the bottom of the window; 
the ray from the top of C would pass through the top of the win- 
dow ; from, the top of i) a little farther oiF^ it would pass through 
the third light; the ray from E would pass through the middle^ 
and F would only occupy the height of one pane. 
19th. Fig. 3. Different sized and shaped objects may produce the 
same image; thus the bent rods Ji and C^ and the straight rods 
E and D would produce the same image^ being placed at different 
distances from the eye^ and all contained in the same angle D. S. 
E. As the bent rods ^ and (7 are viewed edgewise they would 
form the same shaped image as if they were straight. The angle 
formed by the rays of light passing from the top and bottom of 
an object to the eye^ as D. S. E, is called the visual angle, and 
the object is said to subtend an angle of so many degrees, measur- 
ing the angle formed at S. 
20th. Of foreshortenijvg. When an object is viewed obHquely 
it appears much shorter than if its side is directly in front of the 
eye ; if for instance we hold a pencil sidewise at arms length op- 
posite the eye, we should see its entire length ; then if we incline 
the pencil a little, the side will appear shorter, and one of the 
ends could also be seen, and the more the pencil is inclined the 
smaller will be the angle subtended by its side, until nothing but the 
end would be visible. Again if a tcheel be placed perpendicular- 
ly opposite the eye, its rim and hub would shew perfect circles, 
and the spokes would all appear to be of the same length, but if we 
incline the wheel a little, the circles will appear to be ellipses, 
and the spokes appear of different lengths, dependant on the an- 



Plate 42. 



FERSPECTTVE . 



Fig. 2 . 



Hrj.J. 




D ei^ A 



/u/. :j. 



T 






W 



^^ 



F7Mte 43. 



PERSPECTIVE 



Fig.l. 



F 




- 


V 


^ 










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^^ 


^Ap. / 




E 


^"-^"'^^^^ ^ 


.^-^^"^X^ 


B 


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G 



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D 




s 








1) 




'-,'-- 


Ji, 


■■'i ' 


M 


\^'- 










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A 








9 ; 



PLATE XLII. 



85 



gle at which they are viewed; the more the wheel is inclined the 
shorter will be the conjugate diameter of the ellipsis, until the 
whole would form a straight Hne whose length would be equal 
to the diameter, and its breadth equal to the thickness of the 
wheel. This decrease of the angle subtended by an object, when 
viewed obliquely, is cdiiled foreshortening. 



PLATE XLIII. 



Figure 1. 



21st. If we suppose a person to be standing on level ground, with 
his eye at *S, the line Jl. F parallel to the surface and about five 
feet above it, and the surface G. E to be divided off into spaces 
of five feet, as at B, C. D and E, then if from >S, with a radius S. 
G, we describe the arc A. G, and from the points B. C. D and 
Ewe draw lines to S, cutting the arc in H. K. L and M, the 
distances between the lines on the arc, will represent the angle 
subtended in the eye by each space, and if we adopt the usual 
mode for measuring an angle, and divide the quadrant into 90°, 
it will be perceived that the first space of five feet subtends an 
angle of 45°, equal to one-half of the angle that would be sub- 
tended by a plane that would extend to the extreme limits of vi- 
sion; the next space from B to C subtends an angle of about 
18 1-2°, from C to D about 8°, and from D to E about 4 1-2°, and 
the angle subtended would constantly become less, until the divi- 
sions of the spaces would at a short distance appear to touch each 
other, a space of five feet subtending an angle so small, that the eye 
could not appreciate it. It is this foreshortening that enables us 
in some measure to judge of distance. 

22nd. If instead of a level plane, the person at S be standing at the 
foot of a hill, the surface being less inclined would diminish less 
rapidly, but if on the contrary he be standing on the brow of a 
hill looking downward, it would diminish more rapidly* honco we 
derive the following axiom: Thk dkckee of forestoktenino 

OF OHJECTS DEPENDS ON THE ANGLE AT WHICH THKY ARE 
VIEWED. 



86 PLATE XLIII. 

23rd. Perspective may be divided into two branches^ linear 
and AERIAL. 

24th. Linear perspective teaches the mode of drawing the lines 
of a picture so as to convey to the eye the apparent shape or 
FIGURE of each object from the point at which it is viewed. 

25th. Aerial perspective teaches the mode of arranging the 
direct and reflected lights^ shades and shadows of a picture^ 
so as to give to each part its requisite degree of tone and color^ 
diminishing the strength of each tint as the objects recede^ until 
in the extreme distance^ the whole assumes a bluish gray which is 
the color of the atmosphere. This branch of the art is requisite 
to the artist who would paint a landscape, and can be better learnt 
by the study of nature and the paintings of good masters, than by 
any series of rules which would require to be constantly varied. 

26th. Linear perspective, on the contrary, is capable of strict mathe- 
matical demonstration, and its rules must be positively followed to 
produce the true figure of an object. 



DEFINITIONS. 

27th. The perspective plane is the surface on which the pic- 
ture is drawn, and is supposed to be placed in a vertical position 
between the spectator and the object — thus in fig. 1, Plate 42, the 
parallelogram jY, 0. P. Q is the perspective plane. 

28th. The ground line or base line of a picture is the seat of 
the perspective plane, as the line Q. P, fig. 1, Plate 42, and 
G. X, fig. 2, Plate 43. 

29th. The Horizon. The natural horizon is the fine in which the 
earth and sky, or sea and sky appear to meet; the horizon in a 
perspective drawing is at the height of the eye of the spectator. If 
the object viewed be on level ground, the horizon will be about 
five feet or five and a half feet above the ground line, as it is repre- 
sented by V. Ly fig. 2. If the spectator be viewing the object 
from an eminence the horizon will be higher, and if the spectator 
be lower than the ground on which the object stands, the horizon 
will be lower; thus the horizon in perspective, means the height 
of the eye of the spectator, and as an object may be viewed by a 
person inclining on the ground, or standing upright on the 
ground, or he may be elevated on a chair or table, it follows that 
the horizon may be made higher or lower, at the pleasure of the 



PLATE XLIII. 



87 



draughtsman ; but in a mechanical or architectural view of a de- 
sign, it should be placed about five feet above the ground line. 

Note. — The tops of all horizontal objects that are below the horizon, and 
the under sides of objects above the horizon, will appear more or less dis- 
played as they recede from or approach to the horizon. 

30th. The station point, or point of view is the position of the 
spectator when viewing the object or picture. 

31st. The point of sight. If the spectator standing at the sta- 
tion point should hold his pencil horizontally at the level of his 
eye in such a position that the end only could be seen, it would 
cover a small part of the object situated in the horizon; this point 
is marked as at S, fi§. 2, and called the point of sight. It must be 
remembered that the point of sight is not the position of a specta- 
tor when viewing an object; but a point in the horizon directly 
opposite the eye of the spectator, and from which point the spec- 
tator may be at a greater or less distance. 

32nd. Points of distance are set off on the horizon on either 
side of the point of sight as at D, D'j and represent the distance 
of the spectator from the perspective plane. As an object may be 
viewed at different distances from the perspective plane, it fol- 
lows that these points may be placed at any distance from the 
point of sight to suit the judgment of the draughtsman, but they 
should never be less than the base of the picture. 

Note 1. — Although the height of the horizon, and the points of distance may 
be varied at pleasure, it is only from that distance and with the eye on a le- 
vel with the horizon that a picture can be viewed correctly. 

Note 2. — In the following diagrams the points of distance have generally 
been placed within the boundary of the plates, as it is important that the 
learner should see the points to which the lines tend; they should be copied 
with the points of distance much farther off. 

33rd. Visual Rays. All lines drawn from the object to the eye 
of the spectator are called visual rays. 

34th. The middle ray, or central visual ray is a line pro- 
ceeding from the eye of the spectator to the point of sight; exter- 
nal visual raijs are the rays proceeding from the opposite sides 
of an object, or from the top and bottom of an object to the eye. 
The angle formed in the eye by the external rays, is called the 
visual angle. 

Note. — The perspective plane must always be perpendicular to the middle 
visual ray. 

35th. Vanishing Points. It has been shewn at fig. 1 in this 
plate that objects of the same size subtend a constantly decreasing 



88 PLATE XLIII. 

angle in the eye as they recede from the spectator^ until they are 
no longer visible; the point where level objects become invisible 
or appear to vanish^ will always be in the horizon^ and is called 
the vanishing point of that object. 
36th. The point of sight is called the principal vanishing 
poiNT^ because all horizontal objects that are parallel to the mid- 
dle visual ray w^ill vanish in that point. If we stand in the mid- 
dle of a street looking directly toward its opposite end as in Plate 
54^ (the Frontispiece^) all horizontal lines^ such as the tops and 
bottoms of the doors and windows, eaves and cornices of the 
houses, tops of chimnies, &:c. will tend toward that point to which 
the eye is directed, and if the lines were continued they would 
unite in that point. Again, if we stand in the middle of a room 
looking towards its opposite end, the joints of the floor, corners 
of ceiling, washboards and the sides of furniture ranged against 
the side walls, or placed parallel to them, would all tend to a point 
in the end of the room at the height of the eye. 
87th. The vanishing points of horizontal objects not parallel 
with the middle ray will be in some point of the horizon, but not 
in the point of sight. These vanishing points are called acciden- 
tal points. 
38th. Diagonals. Lines drawn from the perspective plane to 
the point of distance as JV, D' and 0. Z), or from a ray drawn to 
the point of sight as E. D' and F, jD, are called diagonals; all such 
lines represent lines drawn at an angle of 45° to the perspective 
plane, and form as in this figure the diagonals of a square, whose 
side is parallel to the perspective plane. 
39th. Of VANISHING PLANES. On taking a position in the mid- 
dle of a street as described in paragraph 36, it is there stated that 
all lines will tend to a point in the distance at the height of the 
eye, called the point of sight, or principal vanishing point; this is 
equally true of horizontal or vertical planes that are parallel to the 
middle visual ray : for if we suppose the street between the curb 
stones, and the side walks of the street to be three parallel hori- 
zontal planes as in Plate 54, their boundaries will all tend to 
the vanishing point, until at a distance, depending on the breadth 
of the plane, they become invisible. Again, the walls of the houses 
on both sides of the street are vertical planes, bounded by the 
eaves of the roofs and by their intersection with the horizontal 
planes of the side walks, these boundaries would also tend to the 
same point, and if the row^s of houses were continued to a suffi- 



PLATE XLIII. 



89 



cient distance, these planes would vanish in the same point; if 
the back walls of the houses are parallel to the front, the planes 
formed by them will vanish in the same point, and if any other 
streets should be running parallel to the first, their horizontal and 
vertical planes would all tend to the same point. 

Note. — A bird's eye view of the streets of a town laid out regularly, would 
fully elucidate the truth of the remarks in this paragraph. When the horizon 
of a picture is placed very high above the tops of the houses, as if the spec- 
tator were placed on some very elevated object, or if seen as a bird would 
see it when on the wing, the view is called a bird's eye view; in a represen- 
tation of this kind the tops of all objects are visible, and the tendency of all 
the planes and lines parallel to the middle visual ray to vanish in the point 
of sight, is very obvious. 

40th. If we were viewing a room as described in paragraph 36, 
the ceiling and floor would be horizontal planes, and the walls 
vertical planes, and if extended would all vanish in the point of 
sight; or if we were viewing the section of a house of several 
stories in height, all the floors and ceilings would be horizontal 
planes, and all the parallel partitions and walls would be vertical 
planes, and would all vanish in the same point. 

41st. When the boundaries of inclined planes are horizontal 
lines parallel to the middle ray, the planes will vanish in the point 
of sight; thus the roofs of the houses in Plate 54, bounded by 
the horizontal lines of the eaves and ridge, are inclined planes 
vanishing in the point of sight. 

42nd. Planes parallel to the plane of the picture have 
no vanishing point, neither have any lines drawn on such planes. 

43rd. Vertical or horizontal parallel planes runnins: at 
any inclination to the middle ray or perspective plane, vanish in 
accidental points in the horizon, as stated in paragraph 37 ; as for 
example, the walls and bed of a street running diagonally to the 
plane of the picture, or of a single house as in Plate 53, where the 
opposite sides vanish in accidental points at different distances from 
the point of sight, because the walls form different angles with the 
perspective plane, as shewn by the plan of the walls B. D and D. 
C, fig. 1. 

44th. All horizontal lines drawn on a plane, or runnini^ 
parallel to a plane, vanish in the same point as the ]>lane itself. 

45th. Inclined lines vanish in points perpendicularly above or 
below the vanishing point of the plane, and if they form the same 
angle with the horizon in diff^ercnt directions as the 2;ables of the 



12 



90 PLATE XLIII. 

house in fig. 2^ Plate 53; the vanishing points will be equidistant 

from the horizon. 
From what has been said we derive the following axioms; their 

importance should induce the student to fix them well in his 

memory : 

1st. The ANGLE OF REFLECTION OF LIGHT is equal to the angle 

of incidence. See paragraph No. 3^ page 81. 
2nd. The shadow of an object is always darker than the object 

itself. See paragraph 10^ page 82. 
3rd. The degree of foreshortening of objects depends on the 

angle at which they are viewed. See paragraph 20^ page 84. 
4th. The apparent size of an object decreases exactly as its dis- 
tance from the spectator is increased. See paragraph 35^ p. 87. 
5th. Parallel planes and lines vanish to a common point. 

See paragraph 36^ page 88. 
6th. All parallel planes whose boundaries are parallel to the 

middle visual ray^ vanish in the point of sight. See paragraph 

36, page 88. 
7th. All horizontal lines parallel to the middle ray vanish in 

the point of sight. 
8th. Horizontal lines at an angle of 45° with the plane of 

the picture, vanish in the points of distance. See paragraph 38, 

page 88. 
9th. Planes and lines parallel to the plane of the pic- 
ture have no vanishing point. 



PRACTICAL PROBLEMS. 

1st. To draio the perspective representation of the square N. O. P. 
Q, viewed in the direction of the line W. B, with one of its sides 
N. O touching the perspective plane G. L, and parallel with it. 

1st. Draw the horizontal line V. L at the height of the eye. 

2nd. From C, the centre of the side JV. 0, draw a perpendicular 
to F. X, cutting it in S, Then S is the point of sight or the prin- 
cipal vanishing point, and C S the middle visual ray. 

3rd. As the sides JV. P and 0. Q are parallel to the middle ray 
C. S^ they will vanish in the point of sight. Therefore from J^ 
and draw rays to S; these are the external visual rays. 

4th. From »S, set off the points of distance T), D' at pleasure, equi- 
distant from Sy and from JY and 0, draw the diagonals Jf. D' and 



PLATE XLIII. 



91 



0. D. Then the intersection of these diagonals with the external 
visual rays determine the depth of the square. 
5th. Draw E, F parallel to JY, 0. Then the trapezoid JT. 0. E. F 
is the perspective representation of the given square viewed at a 
distance from JV on the line W, B, equal to S, D, 

2nd. To draw the Representation of another Square of the same 
size immediately in the rear q/" E. F. 

1st. From E^ draw E, U^ intersecting 0. S in H^ and from F^ 

draw F, D, intersecting JY. S in B. 
2nd. Draw B. H parallel to E, Fy which completes the second 

square ; and the trapezoid JY, 0, H. B is the representation of a 

parallelogram whose side 0. H is double the side of the given 

square. 

Note. — If from W on the line W. B we set off the distance ^S*. D, extending 
in the example outside of the plate, (which represents the distance from which 
the picture is viewed,) and from JV*and draw rays to the point so set off, 
cutting P. Q'm R and T, then the lines R. T and E. F will be of equal 
length, and prove the correctness of the diagram. 



PLATE XLIV. 



Figure 1. 



To draw a Perspective Plan of a Sqxiare and divide it info a given 
number of Squares^ say sixty four. 

Let G. L be the base line, V. L the horizon, S the point of sighl, 
and JY. the given side of the square. 

1st. From JVand 0, draw rays to *S^ and diagonals to /). /), inter- 
secting each other in P and Q, draw P. Q. 

2nd. Divide JY into eight equal parts, and from the points of 
division draw rays to S. 

3rd. Through the points of intersection formed wiili those rays by 
the diagonals, draw lines parallel to ./V'. O, which will divide the 
square as required, and may represent a checker boanl or a 
pavement of scjuare tiles. 



92 plate xliv. 

Of Half Distance. 



When the points of distance are too far off to be used convenient- 
ly^ half the distance may be used; as for example, if we bisect S. 
D in 1-2 D, and JV, in C^ and draw a line from C to 1-2 D, 
it will intersect JY, S in P, being in the same point as by the diago- 
nal drawn from the opposite side of the square, to the whole 
distance at D. 

Note. — Any other fraction of the distance may be used, provided that the 
divisions on the base line be measured proportionately. 

Figure 2. 



To draio the Plan of a Room with Pilasters at its sides, the base 
line, horizon, point of sight, and points of distance given. 

Note. — To avoid repetitions, in the following diagrams we shall suppose 
the base line, the horizon V. L, the point of sight S, and the points of 
distance D. D to be given. 

1st. Let JY, be the width of the proposed room, then draw JY. 
S and 0. S representing the sides of the room. 

2nd. From JY toward lay down the width of each pilaster, and 
the spaces between them, and draw lines to D, then through the 
points where these lines intersect the external visual ray JY. S, 
draw lines parallel with JY. to the line 0. S. 

8rd. From JY and 0, set off the projection of the pilasters and 
draw rays to the point of sight. The shaded parts shew the posi- 
tion of the pilasters. 

4th. If from JY we lay off the distances and widths of the pilasters 
toward J\f, and draw diagonals to the opposite point of distance, 
JY. S would be intersected in exactly the same points. 

Note. — Any re»'-tangular object may be put in perspective by this method, 
without the necessity of drawing a geometrical plan, as the dimensions may 
all be laid off on the. gi'ound line by any scale of equal parts. 

Figure 3. 



To shorten the depth of a perspective draiving, thereby producing 
the same effect as if the points of distance were removed much 
farther off, 

1st. Let all the principal lines be given as above, and the pilasters 
and spaces laid off on the base line from JY. 



F^jte 44. 



FEHSPECTI l^' . 



Fig.]. 





f'f^/.l. 




r '" MtmJh 



Flate Fo. 



pehspectijt:. 

Tesse/ated Fa vem en ts . 




Fif/. ?. 




B- '-^:--^ 



Fig. 3. 




T 5 1 



W^l^n-Vl^A 



Fdmank^ 



PLATE XLIV. 



93 



2nd. From the dimensions on the base line draw diagonals to the 
point of distance D, The diagonal from M the outside pilaster 
will intersect JV, S in P, 

3rd. From JY erect a perpendicular JV, B to intersect the diago- 
nalsj and from those intersections draw horizontal lines to inter- 
sect j\r. s, 

4th. If from JV we draw the inclined line JV. E and transfer the 
intersections from it to V, 0, it will reduce the depth much more^ 
as shown at 0. *S^. 

Most of the foregoing diagrams may be drawn as well wuth one 
point of distance as with two. 



PLATE XLV. 



TESSELATED PAVEMENTS 



Figure 1. 



To draw a pavement of square tiles ^ with their sides placed diago- 
nally to the perspective plane. 

1st. Draw the perspective square J\*. 0, P. Q, 

2nd. Divide the base line JV, into spaces equal to the diagonal 

of the tiles. 
3rd. From the divisions on JV. draw diagonals to the points of 

distance. 
4th. Tint every alternate square to complete the diagram. 

Figure 2. 



To draw a pavement of square black tiles ivith a ivhite border ai'ound 
themj the sides of the squares parallel to the perspective plane and 
middle visual ray. 

1st. Draw the perspective square, and divide A'. into ahernato 
spaces e([ual to the breadth of the stjuare and borders. 

2nd. From the divisions on X draw rays to the point of sight, 
and from X draw a diagonal to the point of distance. 



94 PLATE XLV. 

3rd. Through the intersections formed by the diagonal^ with the 
rays drawn from the divisions on X. 0, draw lines parallel to X. 
Oj to complete the small squares. 



Figure 3. 



To draw a Pavement composed of Hexagonal and Square Blocks, 

1st. Divide the diameter of one of the proposed hexagons a, h into 
three equal parts^ and from the points of division draw rays to 
the point of sight. 

2nd. From a^ draw a diagonal to the point of distance^ and through 
the intersections draw the parallel lines. 

3rd. From 1, 2^ 3 and 4^ draw diagonals to the opposite points of 
distance^ which complete the hexagon. 

4th. Lay off the base line from a and h into spaces equal to one- 
third of the given hexagon^ and draw rays from them to the point 
of sight; then draw diagonals as in the diagram^ to complete the 
pavement. 



PLATE XLVl. 



Figure 1. 



To draw the Double Square E. F. G. H, vieived diagonally, with 
one of its corners touching the Perspective Plane, 

1st. Prolong the sides of the squares as shewn by the dotted lines 

to intersect the perspective plane. 
2nd. From the points of intersection^ draw diagonals to the points 

of distance^ their intersections form the diagonal squares. 
3rd. The square A. B. JV, is drawn around it on the plan and 

also in perspective^ to shew that the same depth and breadth is 

given to objects by both methods of projection. 



PIrrfi' 16. 



FERSFECTIVE 



Fi^.l. 



S 


L. 
A 


< > 




V 


^ / 

/ 

/ 

/ 



Fig. 2. 




W^- MvnjU^e 



IlZmoji k Sons 



PLATE XLVI. 



95 



Figure 2. 



To draw the Perspective Representation of a Circle viewed directly 
ill front and touching the Perspective Plane, 



Find the position of any number of points in the Curve, 

1st. Circumscribe the circle with a square^ draw the diagonals of 
the square P, and JV. Q, and the diameters of the circle ./^. B 
and E. F, also through the intersections of said diagonals with the 
circumference^ draw the chords R. R, R. R, continued to meet 
the line G. Lin Fand F/ 

2nd. Put the square in perspective as before shewn^ dra\v the 
diagonals A^. D'^ and 0. D^ and the radials F. S and YJ S, 

3rd. From Ay draw A, Sy and through the intersection of the dia- 
gonals draw E. F parallel to Jf, 0. 

4th. Through the points of intersection thus found^ viz: A. B. E. 
F, R. R. R, R trace the curve. 

Note 1. — This method gives eight points through which to trace the curve, 
and as these points are equidistant in the plan, it follows that if the points 
were joined by right lines it would give the perspective representation of an 
octagon; by drawing diameters midw^ay between those already drawn on 
the plan, eight other points in the curve may be found. This would give six- 
teen points in the curve, and render the operation of tracing much more correct. 

Note 2. — A circle in perspective may be considered as a polygon of an in- 
finite number of sides, or as a figure composed of an infinite number of 
points, and as any point in the curve may be found, it follows that every 
point may be found, and each be positively designated by an intersection ; 
in practice of course this is unnecessary, but the student should remember, 
that the more points he can positively designate without confusion, the more 
correct will be the representation. 



PLATE XLVII. 



LINE OF E L E V A V I N 



Figure 2 



Is the i)hui of a square whose side is nine fcet^ each t^ide is divided 
into nine parts, and lines from the divisions drawn across in o|ij)o- 
site directions; the surface is therefore divided intc^ eighly-ont* 
S(|uares. (J. Ly fig. 1, is the base line ami I). I) thc^ horizon. 



96 PLATE XLVII. 

1st. — To put the plan ivith its divisions in perspective^ one of its 
sides X. to coincide with the perspective plane. 

Transfer the measures from the side JY. 0, fig. 2, to jY, on the 
perspective plane fig. 1^ and put the plan in perspective by the 
methods before described. 

2nd. — To erect square pillars mi the squares J\^.,Q. TV, nine feet 
high and one foot diameter , equal to the size of one of the squares 
on the plan. 

1st. Erect indefinite perpendiculars from the corners of the squares. 

2nd. On JY. A one of the perpendiculars that touches the perspec- 
tive plane lay off the height of the column JV. ^1/ from the ac- 
companying scale, then JV. J^I is a line of heights on which 
the true measures of the heights of all objects must be set. 

3rd. Two lines drawn from the top and bottom of an object on the 
line of heights to the point of sight, point of distance, or to any 
other point in the horizon, forms a scale for determining similar 
heights on any part of the perspective plan. To avoid confusion 
they are here drawn to the point B. 

4th. Through M draw M. C, parallel to jY. 0, and from C draw 
a line to the point of sight which determines the height of the 
side of the column, and also of the back column erected on Q, 
and through the intersection of the line C. S with the front per- 
pendicular, draw a horizontal line forming the top of the front 
side of the column Q. 

5th. To determine the height of the pillar at IF, 1st. draw a hori- 
zontal line from its foot intersecting the proportional scale JV. B 
in Y; 2nd. from Y draw a vertical line intersecting M. B in X ; 
then F. X is the height of the front of the column W. By the 
same method the height of the column Q may be determined as 
shewn at R. T. 

3rd. — To draw the Caps on the Pillars. 

1st. On the line C. E s. continuation of the top of the front, set off | 
the amount of projection C. E, and through E draw a ray to the 
point of sight. 

2nd. Through C draw a diagonal to the point of distance, and 
through the point of intersection of the diagonal with the ray 



FlM^ //7. 



LINE OJ^' JXEV^rnOX 



A 



D. It/ 



Hg.l. 




x \ 



Fir/. ?. 















I 


■ 


r' 














IN 












































vV 


H ' 




























/ 






























^^ 


■ 














J 



J L 



S/ti/c ttf /'rr/ 



11"' M,r 



PLATE XLVII. 



97 



last drawn^ draw the horizontal line H forming the lower edge 
of the front of the cap. 

3rd. Through M draw a diagonal to the opposite point of dis- 
tance, which determines the position of the corners H and K, 
from H draw a ray to the point of sight. • 

4th. Erect perpendiculars on all the corners, lay off the height of 
the front, and draw the top parallel with the bottom. A ray from 
the corner to the point of sight, will complete the cap. 

The other caps can be drawn by similar means. 

As 3, pillar is a square column the terms are here used indiscrimi- 
nately. 

4th. — To erect Square Pyramids on O and P of the same height as 
the PilhrSy loith a base of four square feet^ as shewn in the plan, 

1st. Draw diagonals to the plan of the base, and from their inter- 
section at R draw the perpendicular R, T' . 

2nd. From B! draw a line to the proportional scale JV, B, and 
draw the vertical line Z. (j, which is the height of the pyramid. 

3rd. Make R. T' equal to Z. G, and from the corners of the per- 
spective plan draw lines to T', which complete the front pyramid. 

4th. A line drawn from T to the point of sight will determine the 
height of the pyramid at a. 

Note 1. — The point of sight S shewn in front of the column TF, must be 
supposed to be really a long distance behind it, but as we only see the end 
of a line proceeding from the eye to the point of sight, we can only represent 
it by a dot. 

Note 2. — A part of the front column has been omitted for the purpose of 
shewing the perspective sections of the remaining parts, the sides of these 
sections are drawn toward the point of sight, the front and back lines are 
horizontal. The upper section is a little farther removed from the horizon, 
and is consequently a little wider than the lower section. This may be 
taken as an illustration of the note to paragraph 29 on page 87, to which 
the reader is referred. 

Note 3. — The dotted lines on the plan shew the direction and boundaries of 
the shadows ; they have been projected at an angle of 45° with the jdane 
of the picture. 



IT 



98 



PLATE XLVIII. 



Figure 1. 



To draw a Series of Semicircular Arches viewed directly in front, 
forming a Vaulted Passage, loith projecting ribs at intervals, as 
sheivn by the tinted plan below the ground line, 

1st. From the top of the side walls JV. / and 0, K, draw the front 
arch from the centre H, and radiate the joints to its centre. 

2nd. From the centre H and the springing Hnes of the arch^ and 
from the corners A and M draw rays to the point of sight. 

3rd. From A and M set off the projection of the ribs^ and draw 
rays from the points so set off to the point of sight. 

4th. Transfer the measurements of A", B", C", &lc., on the plan to 
A'. B'. C, &LC., on the ground line, and from them draw diago- 
nals to the point of distance^ intersecting the ray A. S in B, C. 

5th. From the points of intersection in A. S draw lines parallel to 
the base Hne to intersect M, S. This gives the perspective plans 
of the ribs. 

6th. Erect perpendiculars from the corners of the plans to inter- 
sect the springing lines^ and through these intersections draw 
horizontal dotted lines^ then the points in which the dotted lines 
intersect the ray drawn from H the centre of the front arch^ will 
be the centres for drawing the other arches; R being the centre 
for describing the front of the first rib. 

7th. The joints in the fronts of the projecting ribs radiate to their 
respective centres^ and the joints in the soffit of the arch radiate 
to the point of. sight. 

Note. — No attempt is made in this diagram to project the shadows, as it 
would render the lines too obscure. But the front of each projection is tint- 
ed to make if more conspicuous. 



1^7 ate 48. 



ARCHES' IN FERSFECTTVE 



Fi^J. 




W"}I£j-Lifu. 



PLATE XLVIII. 



99 



Figure 2. 



To draw Semicircular or Pointed Arcades on either side of the 
spectator J running parallel to the middle visual ray. N. P and 
Q. O the width of the arches being given, and P. Q the space be- 
tween them, 

1st. From JV*. P. Q and erect perpendiculars^ make them all of 
equal length, and draw E, F and M. J. 

2nd. For the semicircular arches, bisect E, F in C, and 
from E. C. F. and Q, draw rays to the point of sight. 

3rd. From C, describe the semicircle E, F, 

4th. Let the arches be the same distance apart as the width Q. 0, 
then from draw a diagonal to the point of distance, cutting Q, 
S in /?, from R draw a diagonal to the opposite point of distance 
cutting 0. S in F, from V draw a diagonal to D, cutting Q. S in 
W, and from W to D', cutting 0. S in X 

5ih. Through R, V,^ ^and X, draw horizontal lines to intersect 
the rays 0. S and Q. S, and on the intersections erect perpen- 
diculars to meet the rays drawn from E and F. 

6th. Connect the tops of the perpendiculars by horizontal lines, 
and from their intersections with the ray drawn from Cin 1, 2, 
3 and 4, describe the retiring arches. 

7th. For the gothic arches, (let them be drawn the same dis- 
tance apart as the semicircular,) continue the horizontal lines 
across from R and F, to intersect the rays P. S and JY, S, and 
from the points of intersection erect perpendiculars to intersect 
the rays drawn from M and J. 

8th. From M and J successively, with a radius M. J, describe the 
front arch, and from if the crown^draw a ray to S; from Jl and 
B with the radius A. B, describe the second arch, and from K 
and jL, describe the third arch. 

Note. — All the arches in this plate are parallel to the plane of the picture, 
and although each succeeding arch is smaller than the arch in front of it, all 
may be described with the compasses. 



100 



PLATE XLIX. 

TO DESCRIBE ARCHES ON A VANISHING PLANE, 

Figure 1. 



The Front Arch A. N. B^ the Base Line G. L^ Horizon D. S, 
Point of Sight S, and Point of Distance D^ being given, 

1st. Draw H, J across the springing line of the arch, and construct 

the parallelogram E. F, /. H. 
2nd. Draw the diagonals H. F and /. E, and a horizontal line K. 

M, through the points where the diagonals intersect the given 

arch. Then H. K, JY, M and J^ are points in the curve which are 

required to be found in each of the lateral arches. 
3rd. From F and B^ draw rays to the point of sight S. Then if 

we suppose the space formed by the triangle B. S, i^ to be a 

plane surface, it will represent the vanishing plane on which the 

arches are to be drawn. 
4th. From B^ set off the distance B. A to Z, and draw rays from 

Z. J and C^ to the point of sight. 
5th. From Z, draw a diagonal to the point of distance, cutting B, 

S'u\ 0; through 0, draw a horizontal line cutting Z. S in P; 

from P, draw a diagonal intersecting B, S m Q ; through Q, 

draw a horizontal line, cutting Z. S in P, and so on for as many 

arches as may be required. 
6th. From 0. Q. S. U, erect perpendiculars, cutting F. S in V, 

W, X F. 
7th. Draw the diagonals J. F, F. /, &c. as shewn in the diagram, 

and from their intersection erect perpendiculars to meet P. S; 

through which point and the intersections of the diagonals with 

C. S trace the curves. 

Figure 2. 



To draw Receding Arches on the Vanishing Plane J. S. D, loith 
Piers between them^ corresponding with the given front vieic, the 
Piers to have a Square Base with a side equal to CD. 

1st. From D on the base line, set off the distances D. C, C. B 



r/y/W 4fJ 



ARCHES nVPKHSrECTIVE. 




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Fuj. :\ 




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h'"' Mtn'Ju- 



PLATE XLIX. 



101 



and B, A\o D, E, E, F ^rnd F, G, and from E. F. G, &c. draw 

diagonals to the point of distance to intersect D. S. 
2nd. From the intersections in D. S^ erect perpendiculars; draw 

the parallelogram M. JV. H. I around the given front arch, the 

diagonals M. I and H. JV, and the horizontal line L. K, prolong 

K Ito J and M. JY to F. 
3rd. From B, C. D, M. /. J, K and V^ draw rays to the point of 

sight, put the parallelograms and diagonals in perspective at 0. P 

V. W and at Q. W, R. X, and draw the curves through the points 

as in the last diagram. 
4th. From i where E. U cuts D. S, draw a horizontal line cutting 

B. S in hy and from h erect a perpendicular cutting M. S in k, 
5th. From Y, the centre of the front arch, draw a ray to the point 

of sight, and from A:, draw a horizontal line intersecting it in Z. 

Then Z is the centre for describing the back line of the arch with 

the distance Z. k for a radius. 

Note. — The backs of the side arches are found by the same method as the 
fronts of those arches. The lines are omitted to avoid confusion. 

The projecting cap in this diagram is constructed in the same manner as the 
caps of the pillars in Plate 47. 



PLATE L. 

APPLICATION OF THE CIRCLE WHEN PARALLEL TO THE 

PLANE OF THE PICTURE. 



F. L is the horizon, and S the point of sight. 
Figure 1 



To draw a Semicircular Thin Band placed above the horizon. 

Let the semicircle Jl. B represent the front edge of the band, .//. 
B the diameter, and C the centre. 
1st. From J. C and B, draw rays to the point of sight. 
2nd. From Cthe centre, lay oil* toward J5, the breadth of the band 

a E. 

3rd. From E, draw a diagonal to the point of distance, intersect- 
ing C. S in F. Then F is the centre for describinp; the back of 
the band. 



102 



PLATE L. 



4th. Through F^ draw a horizontal line intersecting A, S in K, 
and B. S in L. Then F. K or F. L is the radius for describing 
the back of the band. 

Figure 2. 



To draw a Circular Hoop with its side resting on the Horizon, 

The front circle jl, H B, K, diameter A, By and centre C being 

given. 

1st. From A, Cand B, draw rays to the point of sight. 
2nd. From C the centre^ lay off the breadth of the hoop at E. 
3rd. From E, draw a diagonal to D', intersecting- C S in F, and 

through Fy draw a horizontal line intersecting A, S in Ky and B, 

S in L, 
4th. From F with a radius F, L or F. Ky describe the back of the 

curve. 



Figure 3. 



To draw a Cylindrical Tub placed below the Horizony whose dia- 
meter y depth and thickness are given. 

1st. From the centre C describe the concentric circles forming the 

thickness of the tub^ lay off the staves and radiate them toward 

C 
2nd. Proceed as in figs. 1 and 2 to draw rays and a diagonal to 

find the point Fy and from F describe the back circles as before ; 

the hoop may be drawn from Fy by extending the compasses a 

little. 
3rd. Radiate all the lines that form the joints on the sides of the 

tub toward the point of sight. 



Figure 4 



Is a hollow cylinder placed below the horizon^ and must be drawn 
by the same method as the preceding figures ; the letters of re- 
ference are the same. 

Note. — The objects in this Plate are tinted to shew the different surfaces 
more distinctly without attempting to project the shadows. 










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103 



PLATE LI. 



The object and point of view given, to find the Perspective Plane 

and Vanishing Points, 

Rule 1. — The Perspective Plane must be drawn perpendicular 
to the middle visual ray. 

Rule 2. — The Vanishing Point of a line or plane is found by 
drawing a line through the station point parallel with such line 
or plane to intersect the perspective plane. The point in the hori- 
zon immediately over the intersection so found^ is the vanishing 
point of all horizontal hues in said plane^ or on any plane parallel 

to it. 
1st. Let the parallelogram E. F, G. Hhe the plan of an object to 

be put in perspective^ and let Q be the position of the spectator 

viewing it^ (called the point of view or station point,) with the 

eye directed toward K, then Q. K will be the central visual ray, 

and K the point of sight. Draw F, Q and H, Q, these are the 

external visual rays. 

Note.— The student should refer to paragraphs 30 and 31, page 87, for the 
definitions of station point and point of sight. 

2nd. Draw P. at right angles to Q. K, touching the corner of 
the given object at ^, then P. O will be the base of the perspec- 
tive plane. 

Note. — This position of the perspective plane, is the farthest point from the 
spectator at which it can be placed, as the whole of the object viewed must 
be behind it; but it may be placed at any intermediate point nearer the 
spectator parallel with P. 0. 

3rd. Through Q draw Q. P parallel with E. F, intersecting the 
perspective plane in P, then P is the vanishing point of the lines 
E. F and G. H. 

4th. Through Q draw Q. 0, parallel to E. H, intersecting the 
perspective plane in 0, then O is the imnishing point for E. H 
and F. G. 

5th. If we suppose the station point to be removed to •//, then ./:/. 
M will be the central visual ray, .//. F and .//. // the external 
rays, and B. D the perspective plane, B the vanishing point of 



104 PLATE LI. 

E. F and G. H^ and the vajiishing point of E, n and F. G will 

be outside the plate about ten inches distant from A, in the direc- 
tion of A. C. 

6th. If the station point be removed to K^ it will be perceived that 
E. H and F. G will have no vanishing pointy because they are 
perpendicular to the middle ray^ and a Hne drawn through the 
station point parallel with the side E. H will also be parallel with 
the perspective plane^ consequently could never intersect it. 

7th. The sides E. F and G. H of the plan^ would vanish in the 
point of sight^ but if an elevation be drawn on the plan in that 
position which should extend above the horizon^ then neither of 
those sides could be seen^ and the drawing would very nearly 
approach to a geometrical elevation of the same object. 

Note. — In the explanation of this plate, the intersections giving the point of 
sight and vanishing points, are made in the perspective plane, which the 
student will remember when used in this connection, is equivalent to the base 
line or ground line of the picture, being the seat or position of the plane on 
which the drawing is to be made; but we must suppose these points to be 
elevated to the height of the eye of the spectator; in practice, these points 
must be set off on the horizontal line as described in paragraph 32, page 87. 



PLATE Lll. 



To delineate the perspective appearance of a Cube vieived acci- 
dentally and situated beyond the Perspective Plane. 



Figure 1. 



Let A. B. C. D be the plan of the cube^ *S' the station pointy S, T 
the middle visual ray and B, L the base line^ or perspective 
plane. 

1st. Continue the sides of the plan to the perspective plane as 
shewn by the dotted lines, intersecting it in M. E. jY and 0. 

2nd. From the corners of the plan draw rays to the station point, 
intersecting the perspective plane in a. d. b. c. 

3rd. Through S, draw S. F parallel to J. D, and S. G parallel 
to D. C. Then F is the vanishing point for the sides A. D and 
B. C; and G is the vanishing point for the sides A. B and D. C 



OBJECT INCLINED TO THE FLAME OF DELfNEATION 



Fir:/. 7 




li M 



I' .1 ,1 1. X 



UT"' Nmi/lr 



PLATE LII. 



105 



Figure 2. 



4th. Transfer these intersections from B. X^fig. 1, to i?. L, fig. 2^ 
and the vanishing points F and G to the horizon^ as shewn by 
the dotted hnes. 

5th. From E and M, draw lines to the vanishing point G^ and 
from JV and 0, draw hnes to the vanishing point F, Then the 
trapezium A. B. C. D formed by the intersection of these lilies^ is 
the perspective view of the plan of the cube, 

6th. To DRAW THE ELEVATION. At M. E, JY and erect per- 
pendiculars and make them equal to the side of the cube. 

7th. From the tops of these perpendiculars draw lines to the op- 
posite vanishing points as shewn by the dotted Hnes^ their inter- 
section will form another trapezium parallel to the first^ repre- 
senting the top of the cube. 

8th. From JL, D and Cy erect perpendiculars to complete the cube. 

Note. — It is not necessary to erect perpendiculars from all the points of in- 
tersection, to draw the representation, but it is done here to prove that the 
height of an object may be set on any perpendicular erected at the point 
where the plane, or line, or a continuation of a line intersects the perspective 
plane ; one such line of elevation is generally sufficient. 

9th. To draw the figure with one line of heights^ proceed as fol- 
lows: from A. D and C, erect indefinite perpendiculars. 

10th. Make E. i/ equal to the side of the cube^ and from ffdraw 
a line to G^ cutting the perpendiculars from D and C inK and L. 

1 1th. From K, draw a line to F^ cutting A. P in P ; from L^ draw 
a line to F, and from P^ draw a line to G^ which completes the 
figure. 

Note. — The student should observe how the lines and horizontal planes be- 
come diminished as they approach toward the horizon, each successive line 
becoming shorter, and each plane narrower until at the height of the eye, the 
whole of the top would be represented by a straight line. I would here re- 
mark, that it would very materially aid the student in his knowledge of per- 
spective, if he would always make it a rule to analize the parts of every dia- 
gram he draws, observe the changes which take place in the forms of ob- 
jects when placed in different positions on tlie plan, and when they are 
placed above or below the horizon at different distances; this would enable 
him at once to detect a false line, and would also enable him to sketch from 
nature with accuracy. Practice this always until it becomes a habit, and I 
can assure you it will be a source of much gratification. 



14 



106 



PLATE LIII. 

TO DRAW THE PERSPECTIVE VIEW OF A ONE STORY 
COTTAGE, SEEN ACCIDENTALLY. 



Figure 1. 



Let A, B, C. D be the plan of the cottage^ twenty feet by four- 
teen feet^ drawn to the accompanying scale; the shaded parts 
shew the thickness of the walls and position of the openings^ 
the dotted hnes outside parallel with the walls, give the projec- 
tion of the roof, and the square E. F, G. H, the plan of the 
chimney above the roof. 

Let P. L be the perspective plane and S the station point. 

1st. Continue the side B, D to intersect the perspective plane in 
Hj to find the position for a line of heights. 

2nd. From all the corners and jambs on the plan^ draw rays toward 
the station point to intersect the perspective plane. 

3rd. Through S draw a line parallel to the side of the cottage D. 
Cy to intersect the perspective plane in L, This gives the vanish- 
ing point for the ends of the building and for all planes parallel 
to it^ viz : the side of the chimney^ and jambs of the door and 
windows. 

4th. Through S draw a line parallel with B, D, to intersect the 
perspective plane^ which it w^ould do at some distance outside of 
the plate ; this intersection would be the vanishing point for the 
sides of the cottage^ for the tops and bottoms of the windows^ 
the ridge and eaves of the roof^ and for the front of the chimney. 

Figure 2. 



Let us suppose the parallelogram P. L. W. X. to he a separate piece 
of paper laid on the other ^ its top edge coinciding with the per- 
spective plane of fig, ly and its bottom edge W. X to be the base 
of the picture^ then proceed as follows ; 

1st. Draw the horizontal line R. T parallel to JV, X and five feet 
above it. 



TIate 33. 



PLJJSr AXB PERSPECTIVE VlEJr 




PLATE LIII. 



107 



2nd. Draw H, K perpendicular to P. L for a line of heights. 

3rd. Draw a line from K to the vanishing point without the pic- 
ture^ which we will call Z ; this will represent the line H. B of 
fig. 1^ continued indefinitely. 

4th. From h and d draw perpendiculars to intersect the last line 
draw n^ in o and e^ which will determine the perspective length of 
the front of the house. 

5th. On K, H set off" twelve feet the height of the walls^ at 0^ and 
from draw a fine to the vanishing point Z^ intersecting d. e in 
m and h. om.n, 

6th. From m and e draw vanishing lines to T^ and a perpendicu- 
lar' from c intersecting them in Y and s ; this will give the cor- 
ner Yy and determine the depth of the building, 

7th. Find the centre of the vanishing plane representing the end^ 
by drawing the diagonals m. Y and e, s^ and through their inter- 
section draw an indefinite perpendicular u. Vy which will give the 
position of the gable, 

8th. To FIND THE HEIGHT OF THE GABLE, SCt ofF itS propOSCd 

height, say 7' 0"^ from to JYon the fine of heights, from JY draw 
a ray to Z, intersecting e, d in W^ and from W draw a vanishing 
line to T intersecting u. v in v^ then v is the peak of the gable. 
9th. Join m. v^ and prolong it to meet a perpendicular drawn 
through the vanishing point J*, which it will do in F, then V is the 
vanishing point for the inclined lines of the ends of the front half 
of the roof The ends of the back of the gables will vanish in a 
point perpendicularly below F, as much below the horizon as V 
is above it. 

10th. For the Roof. Through v draw v. y to Z without, to 
form the ridge of the roof, from /let fall a perpendicular to inter- 
sect y. V in Wy through to draw a fine to the vanishing point V to 
form the edge of the roof. From d let fall a perpendicular to in- 
tersect V. Wy and from the point of intersection draw a line to Z to : 
form the front edge of the roof, from a let fall a perpendicular to de- 
fine the corner Xy and from x draw a line to V intersecting lo. y in 
y, which completes the front half of the roof; from w draw a line 
to the vanishing point below the horizon, from c let lall a perpen- 
dicular to intersect it in g, and through g draw a line to Z, which 
completes the roof. 

11th. For the Chimney. Set off its height above the ridge at 
My from M draw a line toward the vanishing point Z, intersect- 
ing 0. h in Uy from U draw a line to the vanishing point 7', which 



108 PLATE LIII. 

gives the height of the chimney^ bring down perpendiculars from 
rays drawn from G. F and E^ fig. 1^ and complete the chimney 
by vanishing lines drawn for the front toward Z and for the side 
toward T, 

12th For the Door and Windows. Set off their heights at 
P. Q and draw Hnes toward Z^ bring down perpendiculars 
from the rays as before^ to intersect the lines drawn toward Z; 
these lines will determine the breadth of the openings. The 
breadth of the jambs are found by letting fall perpendiculars from 
the points of intersection^ the top and bottom lines of the jambs 
are drawn toward T. 

Note 1. — As the bottom of the front fence if continued, would intersect the 
base line at K the foot of the Une of heights, and its top is in the horizon, it 
is therefore five feet high. 

Note 2. — The whole of the lines in this diagram have been projected accord- 
ing to the rules, to explain to the learner the methods of doing so, and it will 
be necessary for him to do so until he is perfectly familiar with the subject. 
But if he will follow the rule laid down at the end of the description of the last 
plate, he will soon be enabled to complete his drawing by hand, after pro- 
jecting the principal lines, but it should not be attempted too early, as it 
will beget a careless method of drawing, and prevent him from acquiring a 
correct judgment of proportions. 



PLATE LIV. 



FRONTISPIE CE 

Is a perspective view of a street 60 feet wide^ as viewed by a per- 
son standing in the middle of the street at a distance of 134 feet 
from the perspective plane^ and at an elevation of 20 feet from 
the ground to the height of the eye. The horizon is placed high 
for the purpose of shewing the roofs of the two story dwellings. 

The dimensions of the different parts are as follows: 

1st. — Distances across the Picture. 

Centre street between the houses 60 feet wide. 

Side walks, each 10 " 
Middle space between the lines of railway 4 6 ^^ 
Width between the rails 4 9 ^^ 



PLATE LIV. 



109 



Depth of three story warehouse 40 feet. 

Depth of yard in the rear of warehouse 20 " 
Depth of two story dwelling on the right 30 " 

Distances from the Spectator^ in the Line of the 

Middle Visual Ray. 

From spectator to plane of the picture 
From plane of picture to the corner of buildings 
Front of each house 
Front of block of 7 houses 20 feet each 
Breadth of street running across between the blocks 
Depth of second block same as the first 
Depth of houses on the left of the picture^ behind > 
the three story warehouses 5 

To Draw the Picture. 



134 feet 


50 




20 




140 




60 




140 





40 



iC 



1st. Let Cbe the centre of the perspective plane^ H. L the hori- 
zon^ S the point of sight. 

2nd. From C on the line P. P, lay off the breadth of the street 
thirty feet on each side^ at and 60^ making sixty feet^ and from 
those points draw rays to the point of sight; these give the hues 
of the fronts of the houses. 

3rd. From lay off a point 50 feet on P, P, and draw a diagonal 
from that point to the point of distance without the picture ; the 
intersection of that diagonal with the ray from 0, determines the 
corner of the building* from the point of intersection erect a per- 
pendicular to ^. * 

4th. From 50, lay off spaces of 20 feet each at 70, 90 and so on, 
and from the points so laid off draw diagonals to determine by 
their intersection with the ray from 0, the depth of each house. 

5th. After the depth on 0. S is found for three houses, the depths 
of the others may be found by drawing diagonals to the oppo- 
site point of distance to intersect the ray 60 S, as shewn by the 
dotted lines. 

Note. — As a diagonal drawn to the point of distance forms an angle of 45^ 
with the plane of the picture, it follows that a diagonal drawn from a ray to 
another parallel ray, will intercept on that ray a space equal to tlie distance 
between them. Therefore as the street in the diagram is (>0 feet wide and 
the front of each house is 20 feet, it follows that a diagonal drawn from one 
side of tlie street to the other will intercept a space equal to the fronts of 
three houses, as shewn in the drawing. 

6th. Lay off the dimensions on the perspective plane of the 



110 • PLATE LIV. 

depth of the houses^ and the position of the openings on the 
side of the warehouse^ and draw rays to the point of sight as 
shewn by the dotted Unes. 

7th. At erect a perpendicular to D for a Hne of heights ; on this 
Hne all the heights must be laid off to the same scale as the mea- 
sures on the perspective plane^ and from the points so marked 
draw rays to the point of sight to intersect the corner of the 
building at B. For example^ the height of the gable of the ware- 
house is marked at Jty from A draw a ray toward the point of 
sight intersecting the corner perpendicular at B ; then from B, 
draw a horizontal line to the peak of the gable ; the dotted lines 
shew the position of the other heights. 

8th. To find the position of the peaks of the gables on the houses 
in the rear of the warehouses^ draw rays from the top and bottom 
corner of the front wall to the point of sight^ draw the diagonals 
as shewn by the dotted hnes^ and from their intersection erect a 
perpendicular^ which gives the position of the peak^ the intersec- 
tion of diagonals in this manner will always determine the perspec- 
tive centre of a vanishing plane. The height may be laid off on 
0. D at Dj and a ray drawn to the point of sight intersecting the 
corner perpendicular at C, then a parallel be drawn from C to 
intersect a perpendicular from the front corner of the building 
at Ey and from that intersection draw a ray to the point of sight. 
The intersection of this ray^ with the indefinite perpendicular 
erected from the intersection of the diagonals^ will determine the 
perspective height of the peak. 

9th. The front edges of the gables will vanish in a point perpen- 
dicularly above the point of sight, and the back edges in a point 
perpendicularly below it and equidistant. 

10th. As all the planes shewn in this picture except those parallel 
with the plane of the picture are parallel to the middle visual 
ray^ all horizontal lines on any of them must vanish in the point 
of sight, and inchned lines in a perpendicular above or below it, 
as shewn by the gables. 



Ill 



SHADOWS. 



1st. The quantity of light reflected from the surface of an object^ 
enables us to judge of its distance^ and also of its form and posi- 
tion. 

2nd. On referring to paragraph 9^ page 82^ it will be found that 
light is generally considered in three degrees^ viz : light, shade and 
shadow; the parts exposed to the direct rays being in light^ the 
parts inclined from the direct rays are said to be in shade^ and 
objects are said to be in shadow, when the direct rays of light are 
intercepted by some opaque substance being interposed between 
the source of hght and the object. 

3rd. The form of the shadow depends on the form and posi- 
tion of the object from which it is cast, modified by the form and 
position of the surface on which it is projected. For example, if 
the shadow of a cone be projected by rays perpendicular to its 
axis, on a plane parallel to its axis, the boundaries of the shadow 
would be a triangle; if the cone be turned so that its axis would 
be parallel with the ray, its shadow would be a circle; if the cone 
be retained in its position, and the plane on which it is projected 
be inclined in either direction, the shadow would be an ellipsis, 
the greater the obliquity of the plane of projection, the more 
elongated would be the transverse axis of the ellipsis. 

4th. Shadows of the same form may be cast by differ- 
ent figures: for example, a sphere and a flat circular disk 
would each project a circle on a plane perpendicular to the rays 
of light, so also would a cone and a cylinder with their axes par- 
allel to the rays. The sphere would cast the same shadow if 
turned in any direction, but the flat disk if placed edgeways to 
the rays, would project a straight line, whose length would be 
equal to the diameter of the disk and its breadth equal to the 
thickness; the shadow of the cone if placed sideways to the rays 
would be a triangle, and of the cylinder would be a parallelogram. 

5th. Shadows of regular figures if projected on a j^lane retain 
in some degree the outline of the object casting them, more or 



112 

less distorted^ according to the position of the plane ; but if cast 
upon a broken or rough surface the shadow will be irregular. 

6th. Shadows projected from angular objects are generally strong- 
ly defined^ and the shading of such objects is strongly contrasted ; 
thus if you refer to the cottage on Plate 53^ you will perceive 
that the vertical walls of the front and chimney are in light, fully 
exposed to the direct rays of the sun^ while the end of the cot- 
tage and side of the chimney are in shade, being turned away 
from the direct rays^ the plane of the roof is not so bright as the 
vertical walls^, because although it is exposed to the direct rays of 
light it reflects them at a different angle, the shadow of the pro- 
jecting eaves of the roof on the vertical wall forms a dark un- 
broken hne^ the edge of the roof being straight and the surface 
of the front a smooth plane^ the under side of the projecting end 
of the roof is lighter than the vertical wall because it is so situated 
as to receive a larger proportion of reflected light. 

7th. Shadows projected from circular objects are also generally 
well defined; but the shadings instead of being marked by broad 
bold lines as they are in rectangular figures, gradually increase 
from bright light to the darkest shade and again recede as the 
opposite side is modified by the reflections from surrounding ob- 
jects, so gradually does the change take place that it is difficult 
to define the exact spot w^here the shade commences, the fights 
and shades appear to melt into each other, and by its beautifully 
swelling contour enables u& at a glance to define the shape of the 
object. 

8th. Double Shadows. — Objects in the interior of buildings fre- 
quendy cast two or more shadows in opposite directions, as they 
receive the light from opposite sides of the building; this effect is 
also often produced in the open air by the reflected light thrown 
from some bright surface, in this case however, the shadow from 
the direct rays is always the strongest; in a room at night fit by 
artificial means, each light projects a separate shadow, the strength 
of each depending on the intensity of the light from which it is 
cast, and its distance from the object; the student may derive 
much information from observing the shading and shadows of 
objects from artificial light, as he can vary the angle, object and 
plane of projection at pleasure. 

9th. The extent of a shadow depends on the angle of the rays of 
light. If we have a given object and plane on which it is pro- 
jected, its shadow under a clear sky will vary every hour of the 



Flate 55. 
SHADOWS. 



Fhcf.l. 



Ftcf. 9.. 




Fz^. J 



Ficf. 



o 



/ 


__i:_ 




G 




Q 







O 




o 





^ 


o 



-X 



Fz^. 4 





'i/,^'-M7.nzne_ 



lUmnn kSon; 



PLATE LV. 



113 



day^ the sun's rays striking objects m a more slanting position in 
the morning and evening than at noon^ projects much longer 
shadows. But in mechanical or architectural drawings made in 
elevation^ plan or section^ the shadows should always be project- 
ed at an angle of 45°, that is to say^ the depth of the shadow 
should always be .equal to the breadth of the projection or inden- 
tation; if this rule is strictly followed^ it will enable the work- 
man to apply his dividers and scale^ and ascertain his projections 
correctly from a single drawing. 

Note. — The best method for drawing lines at this angle, is to use with the 
T square, a right angled triangle with equal sides, the hypothenuse will be at 
an angle of 45° with the sides ; with the hypothenuse placed against the edge 
of the square, lines may be drawn at the required angle on either side. 



PLATE LV. 



PRACTICAL EXAMPLES FOR THE PROJECTION OF SHADOWS. 



Figure 1 



Is a square shelf supported by two square bearers projecting from 
a wall. The surface of the paper to represent the wall in all the 
folloiving diagrams. 

1st. Let A, B. C. D be the plan of the shelf; A, B its projection 
from the line of the wall W, X; B. D the length of the front of 
the shelf^ and Esmd F the plans of the rectangular bearers. 

2nd. Let G. H be the elevation of the shelf shewing its edge, and 
J and K the ends of the bearers. 

3rd. From all the projecting corners on the plan, chaw lines at an 
angle of 45° to intersect the line of the wall JV. X, and from 
those intersections erect indefinite perpendiculars. 

4th. From all the projecting corners on the elevation, draw lines at 
an angle of 45° to intersect the perpendiculars from correspond- 
ing points in the plan; the points anil lines of intersection ilcline 
the outline of the sliadow as shewn in the diagram. 

15 



114 



PLATE LV. 



Figure 2 
Is a square Shelf against a wall supported by two square Uprights, 

L. M. JV. is the plan of the shelf, P and Q the plans of the up- 
rights^ R, S the front edge of the shelf; T and V the fronts of the 
uprights. 

1st. From the angles on the plan draw lines at an angle of 45° to 
intersect TV, X, and from the intersections erect perpendiculars. 

2nd. From R and S, draw lines at an angle of 45° to intersect the 
corresponding lines from the plan. 



Figure 3 



Is a Frame with a semicircular head^ nailed against a wallj the 
Frame containing a sunk Panel of the same form. 

1st. Let Jl. B. C. D be the section of the frame and panel across 

the middle^ and F on the elevation of the panel^ the centre from 

which the head of the panel and of the frame is described. 
2nd. From E^ draw a Hne to intersect the face of the panel^ and 

from D to intersect JV, Xy and erect the perpendiculars as shewn 

by the dotted hnes. 
3rd. From JY and JVJ draw Hnes to define the bottom shadow^ 

and at L draw a line at the same angle to touch the curve. 
4th. At the same angle draw F. G, make F. H equal to the depth 

of the panel; and F, G equal to the thickness of the frame. 
5th. From H with the radius F, R^ describe the shadow on the 

panel; and from G with the radius F, Sy describe the shadow of 

the frame. 
Note. — The tangent drawn at L and the curve of the shadow touch the 

edge of the frame in the same spot, but if the proportions were different they 

would not do so ; therefore it is always better to draw the tangent. 

Figure 4 



Is a Circular Stud representing an enlarged view of one of the 
JYail Heads used in the last diagram ^ of which JN". O. V is a sec- 
tion through the middle y and W. X the face of the frame, 

1st. Draw tangents at an angle of 45° on each side of the curve. 
2nd. Through L the centre^ draw L. My and make L, M equal to 
the thickness of the stud. 



PLATE LV. 



115 



3rd. From M^ with the same radius as used in describing the stud_, 
describe the circular boundary of the shadow to meet the two 
tangents^ which completes the outline of the shadow. 

Figure 5 



Is a Square Pillar standing at a short distance in front of the 

wall W. X. 

1st. Let Jl, B, C, D be the plan of the pillar^ and W. X the front 
of the wall^ from Jl. C. D draw lines to W, X, and from their 
intersections erect perpendiculars. 

2nd. Let E, F. G. H be the elevation of the pillar^ from F draw 
F, K. L to intersect the perpendiculars from C and D. 

3rd. Through K^ draw a horizontal line^ which completes the out- 
line. The dotted lines shew the position of the shadow on the 
wall behind the pillar. 



PLATE LVI. 



SHADOW S— C N T I N U E D 



Figure 1 



Is the Elevation and Fig, 2 the Flan of a Flight of Steps imth 
rectangular Blockings at the ends^ the edge of the top step even with 
the face of the wall. 

1st. From J.. B. Cand D^ draw Hnes at an angle of 45°. 

2nd. From F where the ray from C intersects the edge of the front 
step, draw a perpendicular to JVT, which defines the shadow on 
the first riser. 

3rd. From Q where the ray from C intersects the edge of the se- 
cond step, draw a perpendicular to Mj which defines the shadow 
on the second riser. 

4th. From if where the ray from Jl intersects the top of the third 
step, draw a peri)endicular to O, which defines the shadow on the 
top of that step. 



116 



PLATE LVI. 



5th. From L where the ray from A intersects the top of the second 
step^ draw a perpendicular to H intersecting the ray drawn from 
C in H, which defines the shadow on the top of the second step. 

6th. From P where the ray from B intersects the ground line^ 
draw a perpendicular to intersect the ray drawn from D in E; 
this defines the shape of the shadow on the ground. 



Figure 3. 



To draw the Shadow of a Cylinder upon a Vertical Plane, 

Rule. — Find the position of the shadow at any number of points. 

1st. From A where the tangental ray (at an angle of 45°) touches 
the plan^ draw the ray to W, X, and from the intersection erect a 
perpendicular. 

2nd. From Jl erect a perpendicular to B^ and from B draw a ray 
at 45° with A. B to intersect the perpendicular from A in L. 
This defines the straight part of the shadow. 

3rd. From any number of points in the plan E. H, draw rays to 
intersect the wall line TV. X, and from those points of intersec- 
tion erect perpendiculars. 

4th. From the same points in the plan erect perpendiculars to the 
top of the cylinder^ and from the ends of these perpendiculars 
draw rays at 45° to meet the perpendiculars on the wall Hne ; the 
intersections give points in the curve. 

Note 1. — The outlines of shadows should be marked by faint lines, and the 
shadow put on by several successive coats of India ink. The student should 
practice at first with very thin color, always keep the camel hair pencil full, 
and never allow the edges to dry until the whole shadow is covered. The 
same rule will apply in shading circular objects ; first wash in all the shaded 
parts w^ith a light tint, and deepen each part by successive layers, always 
taking care to cover with a tint all the parts of the object that require that 
tint; by this means you will avoid harsh outlines and transitions, and give 
your drawing a soft agreeable appearance. 

Note 2. — The lightest part of a circular object is where a tangent to the 
curve is perpendicular to the ray as at P. The darkest part is at the point 
where the ray is tangental to the curve as at A^ because the surface beyond 
that point receives more or less reflected light from surrounding objects. 



Flate 66. 
SHADOWS. 




Fn/. I. 





/'Y.J ■/ 




/v,/. ; 



/■('■'" A// ,•////,• 



INDEX. 






cc 






cc 



Abscissa, 

Absorption of light, 
Accidental points, 
Aerial perspective. 
Altitude of a triangle, 
Angles described. 
Angle of incidence, 
" Visual 

" How to draw angles of 45°, 
Apex of a pyramid, 

" of a cone. 
Application of the rule of 3, 4 and 5, 
Apparent size of an object, . 
Architrave, 
Arc of a circle. 
Arcades in perspective. 
Arches — Composition of 
" Construction of 
" Definitions of 
Arch — Thrust of an 

" Amount of the thrust of an arch, (note) 
Straight arch or plat band. 
Rampant 

Simple and complex arches, 
Names of arches. 
Arches in perspective, . 
Arithmetical perspective, 
Aspect of a country house. 
Axis of a pyramid, 
of a sphere, 
of a cylinder. 
Major and minor axes, 
of a cone. 
Difference between the length of the axes in the sections 

and cylinder, 
of the parabola, 
Axioms in })C'rspc('tive, 
Back of an arch or Kxtrados, 
Band, listel or fillet. 
Base of a triangle. 



PAGE. 

44 
82 
89 
86 
10 
8 
81 
84 
. 113 
39 
44 
17 
90 

m 

12 

99 
54 
54 
54 
55 
55 
55 
55 
55 
55 
98 to 100 
83 
61 
39 
41 
42 
43,44 
45 



of the ciMir 



46 
50 
90 
55 
6S 
9 



118 






PAGE. 


Base of a pyramid, ...... 


39 


" of a cone, ..... 


44 


'' of a Doric column, ..... 


68 


" line or ground line, .... 


86 


Bead described, . . .... 


68 


Bed of an arch, ..... 


55 


Bisect — To bisect a right line, 


14 


" To bisect an angle, .... 


18 


Bird^s eye view, ...... 


89 


Cavetto, a Roman moulding, .... 


69 


Centre of a circle, ...... 


11 


" of a sphere, . . . . 


41 


" of a vanishing plane, .... 


. 110 


Chords defined, ..... 


12 


" Scale of chords constructed, . . . . 


26 


" Application of the scale of chords, 


26 


Circle described, . . . . . . 


11 


" To find the centre of a circle. 


22 


" To draw a circle through three given points, . 


23 


" To find the centre for describing a flat segment, . 


23 


" To find a right line equal to a semicircle. 


23 


" " " " equal to an arc of a circle, 


23 


" Workmen's method of doing the same, 


24 


" Great circle of a sphere. 


41 


" Lesser circle of a sphere, .... 


41 


" Circumferences of circles directly as the diameters, 


74 


" in perspective, ..... 


95 


" Application of the circle in perspective, . 


101 


Circular plan and elevation, .... 


67 


" objects — Shading of . 


112 


Circumference of a circle, . . . . 


11 


" " directly as its diameter. 


74 


Circular domes — To draw the covering of . . . 


52 


Color — How to color shadows, &c. 


116 


Conjugate axis or diameter, .... 


. 43, 44 


" of a diameter of the ellipsis, . 


44 


Contents of a triangle, ..... 


10 


" of a cube, ..... 


37 


" of the surface of a cube, .... 


38 


Complement of an arc or angle. 


13 


Complex and simple arches, .... 


55 


Cone, right, oblique and scalene. 


40 


'' To draw the covering of a cone. 


40 


" Sections of the cone. 


45 


Co-sine, . • • 


13 


Co-tangent, ...... 


13 



^ 


119 




PAGE. 


Co-secant, . . 


13 


Construct — To construct a triangle, .... 


17 


" " an angle equal to a given angle, 


18 


" " an equilateral triangle on a given line, . 


19 


'' '^ a square on a given line, . 


20 


*^ " a pentagon on a given line. 


21 


" " a heptagon on a given line, 


22 


*^ " any polygon on a given line, . 


22 


" " a scale of chords . 


26 


*' " the protractor. 


30 


Construction of arches, ..... 


54 


Contrary flexture — Curve of . 


7 


" " Arch of : . . . . 


57 


Cornice and piazza — Effect of the . . 


62 


Cottage in perspective, ..... 


. 106 


Covering of the cube, . . . . 


38 


" " parallelopipedon, .... 


38 


'' " triangular prism, .... 


38 


" " square pyramid. 


39 


*' " hexagonal pyramid, .... 


40 


" " cylinder, . . . . 


40 


" " cone, ...... 


40 


" " sphere, ..... 


41 


" " regular polyhedrons, . . 


42 


" " circular domes, .... 


52 


Crown of an arch. 


55 


" moulding, ...... 


69 


Cuhe or hexahedron, ... 


37 


Cubic measure, ...... 


37 


Cube — To draw the Isometrical .... 


76 


" in perspective, ..... 


. 104 


Cycloid described, ...... 


35 


Cycloidal arches, . . . . 


36 


Cylinder^ •.••... 


40 


" To draw the covering of a . 


40 


" Sections of the 


42,43 


" Right and oblique ..... 


42 


" To find the section of a segment of a cylinder through 


three 


given points, ..... 


51 


" of a locomotive engine, .... 


75 


" in perspective, ..... 


. 10-2 


" Shadow of a . . 


116 


Cyiiia or cyma recta — Roman, .... 


09 


" " Grecian, ..... 


71 


*^ revcrsa, talon or ogee — Roman, 


69 


" " " " Grecian, : 


71 



120 






PAGE. 


Degree defined, ...... 


12 


Depressed arch, . ..... 


59 


Design for a cottage, . . 


60 


" What constitutes a .... . 


61 


Details of cottage, ...... 


65 


Diamond defined, . . . . ... 


10 


Diagonal defined, ...... 


11 


Diagonal lines in perspective, ..... 


88 


Diameter of a circle, ..... 


11 


" of a sphere, ...... 


Al 


" of an ellipsis, ..... 


. 43, 44 


" of the parabola, ...... 


50 


Definitions of lines, ..... 


7 


" of angles, ...... 


8 


" of superficies, ..... 


9 


" of the circle, ..... 


11 


" of solids, 


36 


" of the cylinder, ..... 


42 


" of the cone, . . . . . 


44 


*' of the parabola, ..... 


50 


" of arches, . , . 


55 


*' in perspective, . . . . 


86 


Directrix of the parabola, ..... 


50 


Distance — Points of . 


87 


« Half ...... 


92 


" The quantity of reflected light enables us to judge of .> 


111 


Dodecahedron^ ...... 


42 


Domes — Covering of hemispherical . . . . 


52 


Double shadows, ...... 


. 112 


Echinus^ or Grecian ovolo, ..... 


70 


Effect — A perspective view necessary to shew the eifect of an intended 


improvement, ...... 


61 


Elevation described, . . 


61 


Ellipsis — False ...... 


34 


" the section of a cylinder, . . . . 


. 43 


" To describe an ellipsis with a string . 


43 


" the section of a cone, .... 


45 


" To describe an ellipsis from the cone . 


45 


" To describe an ellipsis by intersections 


46 


" To describe an ellipsis with a trammel 


47 


Elliptic Arch — To draw the joints of an . 


57 


Epicycloid described, ...... 


36 


Equilateral triangle, ..... 


9 


" arch, (Gothic) ..... 


58 


Extrados or back of an arch, .... 


55 


Fillet^ band or listel. 


68 

1 





121 


1 


PACE. 


Focus — Foci of an ellipsis, . . . . . 


43 


" of a parabola, ...... 


50 


Foreshortening J . . 


84 


" The degree of foreshortening depends on the angle 


; at 


which objects are viewed, . 


. 85, 90 


Form of shadows, . . . . . 


111 


Frustrum of a pyramid, . . • . 


39 


" of a cone, . . . . . . 


45 


Globe or sphere, . . . . . 


41 


Gothic arches described, . , 


58 


Grades^ . . . • 


13 


Grecian mouldings, ...... 


70 


Grounds to plinth, &c., . 


66 


Ground line or base line, . « . . . 


86 


Habit of observation, . . . . . 


105 


ii/a//* distance, . . ' . . • 


92 


Height^ rise or versed sine of an arch, . . . , 


54 


Hemisphere^ ....... 


41 


Hexahedron or cube, ...... 


37 


Hexagonal pavement in perspective, .... 


94 


Hipped roof — hipped rafter, . 


64 


Horizontal or level line, ..... 


8 


" covering of domes, . . . . 


53 

1 


Horizon in perspective, ...... 


86 


Horseshoe arch, ....... 


57 


" pointed arch, ..... 


59 


Hyperbola the section of a cone, ..... 


45 


" To describe the hyperbola from the cone. 


46,48 


Hypothenuse, . . . . . . 


9, 16 


" Square of the . 


16 


Icosahedron, ....... 


42 


Inclined lines in Isometrical drawing require a different scale. 


81 


Incidence — The angle of incidence equal to the angle of reflection, 


81 


Inclined lines — Vanishing point of ... . 


89 


Inclined planes — Vanishing point of .... 


90 


Inscribe — To inscribe a circle in a triangle, .... 


19 


" " an octagon in a square, 


20 


" " an equilateral triangle in a circle, 


21 


" " a square in a circle. 


21 


" " a hexagon in a circle, 


21 


" " an octagon in a circle, 


21 


" " a dodecagon in a circle, 


21 


Intrados or soffit of an arch, ..... 


55 


Isometrical drawing, ...... 


76 


" cube, ... 


76 


" circle, ...... 


79 



16 



122 


-• 




PAGE. 


Isometrical circle — To divide the 


80 


Isosceles triangle, . . . . 


9 


Joints of an arch defined, 


. . 55 


" To draw the joints of arches, . 


56 to 59 


Joists — Plan of a floor of joists, . . . 


65 


" Trimmers and trimming 


65 


'' Tail 


65 


Keystone of an arch. 


55 


Lancet Arch — To describe the . 


58 


Light — Objects to be seen must reflect light. 


.81 


" becomes weaker in a duplicate ratio, &c.. 


81 


" Three degrees of . 


82, 111 


Lines — ^Description of . 


7 


Line — To divide a right 


.25 


" To find the length of a curved 


23 


" Workmen's method of doing so, 


24 


Line o/* centres (of wheels,) 


73 


" Pitch line defined, 


73 


*' To draw the pitch line of a pinion to contain 


a definite number of 


teeth, .... 


74 


" Ground or base line. 


86 


" Vanishing point of a line. 


89 


" of elevation in perspective. 


95 


Linear perspective defined, . 


86 


Listel, band or fillet, .... 


/ 68 


Locomotive cylinder. 


75 


Lozenge defined, .... 


10 


Major and minor axes or diameters, . 


. 43, 44 


Measurements to be proved from opposite ends, . 


: . 63 


Measures — Cubic ... 


.37 


" Lineal and superficial 


., . 37 


" of the surface of a cube, 


38 


Middle ray or central visual ray. 


87 


MinuteSy ..... 


12 


Mitre — To find the cut of a 


18 


Moresco or Saracenic arch, . 


.57 


Mouldings — Roman .... 


68 


" Grecian 


70 


Obelisk defined, . . 


39 


Oblique pyramid, 


39 


" cone. 


40, 45 


" cylinder, .... 


.42 


OftZon^ defined, .... 


10 


Octagonal plan and elevation, 


67 


Octahedron, . . . ... 


42 


Ogee Arch or arch of contrary flexure, 


57 





123 






Ogee or cyma reversa — Roman . 

" " " Grecian . 

Optical illusion, . . . ... 

Ordinate of an ellipsis, .... 

Ovals composed of arcs of circles, 

Ovolo — Roman ..... 

" or Echinus — Grecian . ... 

Parallelogram defined, .... 

Parallel lines, ...... 

" ruler, ..... 

" Application of the parallel ruler, 
Parallelopipedon, ..... 

Parabola — To find points in the curve of the 
the section of a cone. 
To describe a parabola from the cone, 
To describe a parabola by tangents, &c. 
To describe a parabola by continued motion, . 
applied to Gothic arches. 
Definitions of the parabola, . 
Parameter defined, ..... 

Pentag07i reduced to a triangle, .... 

" To construct a pentagon on a given line, . 
Perimeter the boundary of polygons. 
Periphery the boundary of a circle. 
Perpendicular lines defined, .... 

" To bisect a line by a perpendicular, 

" To erect a perpendicular, . 

" To let fall a perpendicular, 

Perspective view necessary to shew the effect of a design, 
Essay on perspective. 
Linear and aerial perspective, 
plane, or plane of the picture, 

" must be perpendicular to the middle visual 
plan of a square, .... 

'* of a room with pilasters, 
" To shorten the depth of a perspective plan, 
Tesselated pavements in perspective. 
Double square in perspective, 
Circle in perspective, 
Line of elevation, .... 

Pillars with projecting caps in perspective. 
Pyramids in perspective, 
Arches s(mmi in front, 

" on a vanishing ])lane. 
Application (>r the circU', 
To iind tlie perspective piano, is.c. 



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PAGE. 

69 


. . 




Tl 


• 




77 
44 


• • 


33, 


34 
69 


• • 




TO 
10 


• • 




7 
24 


• • 




25 

38 


• • 




32 

45 






46 
49 






50 
51 






50 
50 






25 
21 






9 
11 






8 
14 


. 14 


15 


,16 




15 


,16 






61 
81 






86 
86 


1 ray, 


90 


87 
,91 
92 
92 
93 






94 
95 


• 




95 
9(1 
97 
9S 






100 
101 






103 



124 



PAGE. 



Perspective view of a cube seen accidentally, . . .104 

" view of a cottage seen accidentally, . . 106 
" view of a street, ..... 108 

Piazza and Cornice — Their effect on the design for a cottage, 62 

Pillars in perspective, . . . . . .96 

Pitch of a wheel, . . . . . .73 

" circle of a wheel, . . . . . .73 

" line of a wheel, ...... 74 

Plan — A horizontal section, . . . . .60 

Plane superficies, ...... 9 

Planes — Vanishing ...... 88 

" Parallel planes vanish to a common point, . . 90 

" parallel to the plane of the picture, . . .90 

" To find the perspective plane, . . . .. ' 103 

Platband or straight arch, . . . . .55 

Platonic figures, ...... 42 

Plinth — Section of parlor plinth, . . . . .66 

Point of intersection, . . . . . 13 

" of contact, . . . . . . . 13 

" Secant point, . . . ' . . . 13 

" of sight, . . . . . . . 87 

" of view or station point, . . . . 87 

" Vanishing points ...... 87 

" Principal vanishing point, .... 88 

" of distance, ...... 87 

Pointed arches in perspective, . . . . 99 

Poles of the sphere, . . . . . .41 

Proportional diameter of a wheel, . . . . 73 

" circle or pitch line, . . . . .73 

Polygons described, ...... 9 

" Table of polygons, . . . . .19 

" Regular and irregular polygons, . . . 19, 20 

Polyhedrons, . . . . ' . . , 37, 42 

Projecting caps in perspective, .... 96 

Protractor — Construction of the protractor, . . . .30 

" Application of the protractor, ... 30 

Prisms, . . . . . . . 38, 39 

Pyramid, . . . . . . . 39 

" in perspective, ...... 97 

Quadrant of a circle, ...... 12 

Quadrangle defined, . . . . . .10 

Quadrilateral defined, . . . . . . 10 

Radius — Radii, . . . . . . .11 

Rafters — Elevation of rafter, . . . . . 65 

" Hip . . . . . . . 64 

Rampant arch, . . . . . . 55 





12.5 




PAGE. 


Rays of light reflected in straight lines, 


83 


" converged in the crystalline lens, . 


83 


Rectangle defined, ..... 


10 


Reduce — To reduce a trapezium to a triangle, 


25 


" To reduce a pentagon to a triangle, 


25 


Reflection of light, ..... 


81 


" The angle of reflection equal to the angle of incidence 


, . 81 


Reflected light enables us to see objects not illuminated by direct 


rays, 83 


Regular triangles, ..... 


9 


" polyhedrons, . . . . . 


37 and 42 


Requisites for a country residence, .... 


61 


Refraction of light, . . . . ' . 


82 


Retina of the eye. 


83 


Rhomb — Rhombus, ...... 


10 


Rhomboid, ...... 


10 


Right angled triangle. 


9 


Right line defined, ..... 


7 


" pyramid, ..... 


39 


" cylinder, ...... 


40 


" cone, ..... 


40, 45 


Rise or versed sine of an arch, .... 


54 


Rise or riser of stairs, . . . . . 


63 


Roof- — Hipped . . . . . x . 


64 


" Section of roof, ...... 


65 


Roman mouldings, ..... 


68 


Rule of 3, 4 and 5, . . . ^ , 


16, 17 


Saracenic or Moresco arch, .... 


57 


Scale of chords, 


26 


Scales of equal parts, ..... 


27 


" Simple and diagonal scales, . . . , 


28, 29 


" Proportional scale in perspective, 


96 


Scalene triangle, . . . 


.9 


Scheme or segment arch, ..... 


56 


Scotia described — Roman, . . . . 


68 


" " Grecian, ..... 


72 


Secant — Secant point, or point of intersection, 


13 


Seconds, ...... 


12 


Sector of a circle, ...... 


12 


Section — a vertical plan, . 


60 


Sections of the cylinder, . . . 


12, 43 


Section " " Uirough three given points, . 


51 


" of the cone, ...... 


45 to 48 


" of the eye, . . , . . 


83 


Serpentine line, ...... 


7 


Segment of a circle, ..... 


12 


" To find the centre for describing a segment, 


23 



126 






PAGE. 


Segment — To find a right line equal to a segment of i 


a circle, . ■ . 23 


" To describe a segment 'with a triangle, 


31 


" To describe a segment by intersections, 


33 


" of a sphere, . . . . 


41 


" or scheme arch, . . . • 


... 56 


Semicircle, ..... 


12 


Semicircular arch, . . . . 


56 


" " in perspective. 


99 


Shade and shadow. 


82, 111 


Shadow always darker than the object, . 


87, 90 


Shadows — Essay on shadows. 


111 


iS'Aac^in^ of circular objects, . . . 


112 


Shadow — Lightest and darkest parts of a 


. 116 


^S^A^— Method of sight, 


81, 83 


" Point of sight, 


87 


Sills of window, .... 


. . 66 


Simple and complex arches. 


... 55 


SinCy 


13 


Skew-hack of an arch. 


56 


Soffit or intrados of an arch. 


55 


Span of an arch, .... 


54 


Sphere — Definitions of the sphere, 


41 


" To draw the covering of a sphere, . 


. ". . .' 41 


Springing line of an arch. 


. . 54 


Square, ...... 


10 


Square corner in a semicircle, , . 


15 


" " by scale of equal parts, 


16 


" of a number, .... 


16 


" of the hypothenuse, . 


. , . 16 


Stairs — To proportion the number of steps of stairs. 


63 


Station point or point of view. 


.37 


Stiles of sash, &c. .... 


66 


Step or tread, . . . . . 


63 


Straight or right line, . 


. ■ 7 


" arch or plat band, . . . . 


55 


St7'eet in perspective, .... 


108 


Subtense or chord, . . . . . 


12 


Summit of an angle, .... 


8 


" of a pyramid, . . . . 


39 


" of a cone, 


40, 44 


Superficies or surface, . . . . 


9 


Supplement of an angle or arc, . 


13 


Table of the names of polygons, 


.19 


" " the angles of polygons, 


. Plate 10' 


Tail joist, . . . . . 


65 


Talon or Ogee — Roman, 


69 





127 




PAGE. 


Talon or Ogee — Grecian, .... 


71 


Tangent defined, . . . 


13 


Teeth of wheels — To draw the teeth of wheels, 


73 


" " Pitch of the " 


73 


" " Depth of the " 


73 


Tesselated pavements in perspective. 


93 


Tetragon defined, . 


10 


Tetrahedron one of the regular solids. 


41 


Torus described, ..... 


68 


Trapezium defined, . . . . . 


10 


" reduced to a triangle. 


25 


Trapezoid defined, . . . . ' . 


10 


Transverse axis or diameter, 


, 42, 43 


Tread or step, . . . ' . 


63 


Trigons or triangles, .... 


9! 


Trisect — To trisect a right angle. 


18 


Trimmers and trimming joists, . . . 


65 


Truncated pyramid, ..... 


39 

1 


" cone, . . 


. • . 45' 


Tudor or four centred arch, .... 


59 


Vanishing points, . . . 


. ' S7 


" Principal vanishing point. 


88 


<* planes, ..... 


88 


Versed sine of an arc, . 


13 


" "or rise of an arch, . . . ^ 


54 


Vertex of a triangle, . . . 


10 


" of a pyramid, . . . 


39 


" of a cone, ..... 


44, 


" of a diameter of the ellipsis, . 


44 


" Principal vertex of a parabola. 


50' 


" of a diameter of the parabola. 


50. 


Vertical or plumb line, ..... 


8 


" coverings of domes, . 


52 


Visual angle, . . . . 


8-1 


" rays, ..... 


• . 87 ' 


Voussoirs of an arch, . . • . . ' . 


55 


Wheel and pinion — Drawing of a . 


72 


" " To proportion the teeth of a . 


74 


Wheel viewed in perspective, . . . • 


ai 


Windows — Details of windows , . . . 


G5, 66 



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